ACKNOWLEDGEMENTS
The author would like to express his sincerest appreciation to his
research advisor and supervisory committee cochairman. Or. A.J. Mehta,
Associate Professor of Coastal and Oceanographic Engineering, for his
continuous guidance and encouragement throughout the course of this
research. Appreciation is also extended for the valuable advise and
suggestions of the supervisory committee chairman. Dr. B.A. Christensen,
Professor of Civil Engineering, as well as for the guidance received
from the other committee members: Dr. B.A. Benedict, Professor of Civil
Engineering; and Dr. J.L. Eades, Associate Professor of Geology.
Sincere thanks are also due to Dr. E. Partheniades, Professor of
Engineering Sciences, whose interest and suggestions were of a great
help to the author.
The author wishes to acknowledge the assistance provided by Dr.
Ranjan Ariathurai to this investigation. He provided a copy of the
finite element solution routine used in this study and gave advice
regarding its use and related experimental research.
Appreciation is extended to Drs. D.G. Bloomquist and P. Nielsen for
their suggestions and providing references on various aspects of this
study.
Special thanks go to Mr. Vernon Sparkman and the staff of the
Coastal Engineering Laboratory for their assistance with the experiments
ii
PREDICTION OF COHESIVE SEDIMENT MOyEKEMT
IN ESTUARIAL WATERS
By
EARL JOSEPH HAYTER
DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVElSTTf OF PWRIDA IN
PARTIAL FULFILLMENT OF THE REOUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1983
performed during this research. The author wishes to thank Ms. Debbie
LaMar for typing this manuscript and Ms. Lillean Pieter for drafting the
figures. In addition, the author thanks Ms. Lucile Lehmann and Ms.
Helen Twedell of the Coastal Engineering Archives for their assistance.
Great appreciation is extended to the Water Resources Division of
the U.S. Geological Survey for their financial support of this research
through the thesis support program. More specifically, the author
expresses his gratitude to Dr. Robert Baker, regional research
hydrologist, and to Drs. Carl Goodwin and Harvey Jobson who served as
thesis support advisors.
Finally, the author thanks his wife Janet for her love, moral
encouragement and patience, and his parents, George and Lois Hayter, for
their love and support.
iii
TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS ii
LIST OF TABLES viii
LIST OF FIGURES ix
ABSTRACT xvii
CHAPTER
I INTRODUCTION . 1
1.1. Estuarial Cohesive Sediment
Dynamics 1
1.2. Sediment Related Problems
in Estuaries 4
1.3. Approach to the Problems 7
1.4. Scope of Investigation 9
II BACKGROUND MATERIAL 12
2.1. Introductory Note 12
2.2. Description and Properties
of Cohesive Sediments 12
2.2.1. Composition 12
2.2.2. Origin 13
2.2.3. Structure 14
2.2.4. Interparti cle Forces 15
2.2.5. Cation Exchange Capacity .... 19
2.2.6. Coagulation 21
2.3. Significance of Important Physical
Factors in Estuarial Sediment
Transport 30
2.3.1. Estuarial Dynamics 30
2.3.2. Sediment Processes 35
III SEDIMENT TRANSPORT MECHANICS 47
3.1. Introductory Note 47
i V
PAGE
3.2. Governing Equations 47
3.2.1. Coordinate System 47
3.2.2. Equations of Motion 47
3. 2.1. a. Continuity , 49
3.2.1.b. Conservation of
Momentum 49
3.2.3. AdvectionDispersion
Equation 51
3.3. Sediment Bed 53
3.3.1. Bed Structure 53
3.3.2. Effect of Salinity on
Bed Structure 64
3.3.3. Bed Schematization 71
3.4. Erosion 75
3.4.1. Previous Investigations 75
3.4.2. Effect of Salinity
on Resuspension 92
3.4.3. Erosion Algorithm 101
3.5. Dispersive Transport 106
3.5.1. Dispersion Mechanisms 106
3.5.2. Dispersion Algorithm 112
3.6. Deposition 116
3.6.1. Previous Investigations 116
3.6.2. Effect of Salinity
on Deposition 133
3.6.3. Deposition Rates 140
3.6.4. Deposition Algorithm 152
3.7. Consolidation 154
3.7.1. Description 154
3.7.2. Consolidation Algorithm 168
IV MODEL DEVELOPMENT 178
4.1. Introductory Note 178
4.2. Review of Previous Models •^^^
4.3. Model Description 181
4.4. Finite Element Formulation 188
4.4.1. Introductory Note 188
4.4.2. Shape Functions 189
4.4.3. Galerkin Weighted
Residual Method 195
4.4.4. Equation Solvers 201
4.5. Convergence and Stability 201
V
PAGE
V MODEL VERIFICATION AND APPLICATION 203
5.1. Introductory Note 203
5.2. Laboratory Experiments 203
5.2.1. Recirculating Flume
Experiments 204
5. 2.1. a. Facilities 204
5.2. l.b. Instrumentation. . . . 207
5. 2. I.e. Procedure 217
5.2.1. d. Results 222
5.2.2. Rotating Annular Flume
Experiment 227
5. 2. 2. a. Facilities 227
5.2.2.b. Instrumentation. . . , 230
5.2.2.C. Procedure 230
5.2.2.d. Results 233
5.2.3. Model Simulations 233
5.3. Simulation of WES Deposition
Experiment 236
5.4. Discussion of Results , 238
5.5. Model Applications 246
5.6. Model Limitations 257
5.7. Model Applicability , 258
5.7.1. Water Quality Problems ..... 258
5.7.2. Sedimentation Management
Problems 259
VI CONCLUSIONS AND RECOMMENDATIONS 260
6.1. Summary and Conclusions 260
6.2. Recommendations for Future
Research 269
APPENDICES
A Derivation of AdvectionDi spersi on
Equation 271
B Coefficient Matrices in the Element
Matrix Differential Equation 286
vi
PAGE
C Computer Program . 289
C.l. Main Program 289
C.2. Subroutines 290
C.3. Flow Chart 296
C. 4. User's Manual „ 308
0 Data Collection and Analysis
Programs 323
D. l. Field Data Collection
Program , 323
D.2. Laboratory Sediment Testing
Program 330
REFERENCES 335
BIOGRAPHICAL SKETCH 349
vi i
LIST OF TABLES
TABLE PAGE
2.1 Properties of Sediment Aggregates (after Krone,
1963) 41
3.1 Principle Factors Controlling Erosion of
Saturated Cohesive Sediment Beds 77
3.2 Cation Concentrations in Processed Sodium
Chloride and Standard Sea Salt. 95
3.3 Variation of Empirical Coefficients in the
Relationship Between Pi^^^) and T^^ 171
4.1 Quadratic Shape Functions 192
4.2 Derivatives of Shape Functions 194
5.1 Chemical Composition of the Tap Water
(after Dixit, 1982) 217
V i i i
LIST OF FIGURES
FIGURE PAGE
1.1 Schematic Representation of Transport and
Shoaling Processes in the Mixing Zone of a
Stratified Estuary (after Mehta and Hayter,
1931) 3
1.2 Interactions of Tidal and Estuarial Sediment
Transport Processes (after Owen, 1977). ...... 8
2.1 Repulsive and Attractive Energy as a Function
of Particle Separation at Three Electrolyte
Concentrations (after van Olphen, 1963) 17
2.2 Net Interaction Energy as a Function of Particle
Separation at High Electrolyte Concentration
(after van Olphen, 1963) 17
2.3 Net Interaction Energy as a Function of Particle
Separation at Intermediate Electrolyte
Concentration (after van Olphen, 1963) 20
2.4 Net Interaction Energy as a Function of Particle
Separation of Low Electrolyte Concentration
(after van Olphen, 1963) 20
2.5 Comparison of the Collision Functions for
Brownian, Shear and Differential Sedimentation
Coagulation (after Hunt, 1980) 25
2.6 Variation of SAR with Salinity (Sea Salt
Concentration) (after Ariathurai, 1974) 27
2.7 CoagulationDispersion Boundary Curves for (a)
Montmorillonite, (b) II lite and (c) Kaolinite at
Three pH Ranges (after Kandiah, 1974) 27
2.8 Monthly Salinity Distributions in the Cumbarjua
Canal, Goa, India; Ebb; Flood
(after Rao etal_., 1976) 31
2.9 Variation in Chloride Concentration in San
Francisco Bay and SacramentoSan Jaoquin Delta
September 1955 (after Orlob et_al_., 1967) 32
ix
FIGURE
PAGE
2.10 Salinity of the Surface Waters of the Pamlico
River Estuary as a Function of the Distance from
the Railroad Bridge in Washington, D.C. (after
Edzwald et al_., 1974) 33
2.11 Computed Longitudinal Salinity Profile in the
Yangtze River Estuary as a Function of the
Distance Downstream from Jiang Zhen Dong for
Two River Discharges (after Huang et al . ,
1980) 33
2.12 A Plot of Raw Viscometer Data Obtained from the
U.S. Army Corps of Engineers Philadelphia
District Sample (after Krone, 1963) 39
2.13 Time and Depth Variation of Suspended Sediment
Concentration in Savannah River Estuary (after
Krone, 1972) 45
2.14 Schematic Representation of the Physical States
of Cohesive Sediment in Estuary Mixing Zone
(after Mehta et_al_., 1982a) 46
3.1 Coordinate System 48
3.2 Measured Red Density Profiles for Thames Mud
for Two Different Consolidation Times (after
Owen, 1970) 57
3.3 Measured Bed Density Profiles for Avonmouth Mud
for Two Different Initial Suspended Sediment
Concentrations (after Owen, 1970) 58
3.4 Measured Bed Density Profiles for Avonmouth Mud
for Different Bed Thicknesses (after Owen,
1970) 59
3.5 Dimension! ess Density Profiles of Mud Beds
(after Thorn and Parsons, 1980) 59
3.5 Variation of Bed Density with Depth for Three
Different Conditions of FlowDeposited Beds
(after Parchure, 1980) 61
3.7 Dimensionless Density Profile of Mersey Mud with
Tdc = 2 hours (after Bain, 1981) 62
3.8 Dimensionless Bed Density Profile of a Mud Bed
(after Thorn, 1981) . 62
3.9 Dimensionless Density Profiles for Kaolinite
Beds with T^c = 2, 5, 11 and 24 hours 63
X
FIGURE
PAGE
3.10 Dimensionless Density Profiles for Kaolinite
Beds with T^^, = 48, 72, 96, 144 and 240 hours ... 63
3.11 Bed Shear Strength Profiles for Kaolinite Beds
(after Parchure, 1980) 65
3.12 Bed Shear Strength Profile for a Kaolinite Bed
(after Dixit, 1982) 66
3.13 Dimensionless Bed Density Profiles for Salinities
of 0, 1, 2, 5 and 10 ppt 69
3.14 Bed Shear Strength Profiles as Functions of
Salinity 70
3.15 Bed Schematization Used in Bed • Formation
Algorithm 72
3.16 Hypothetical Shear Strength Profile Illustrating
Determination of Bed Layers Thicknesses 72
3.17 Laboratory Determined Relationship Between
Erosion Rate, e, and Bed Shear Stress, t:^
(after Mehta, 1981) 80
3.18 Example of Relationship Between £ and Stress Stress
(after Mehta, 1981) 80
3.19 £  ^^^^ °f Partheniades (1962), Series I
and II (after Mehta, 1981) 81
3.20 Dimensionless ^  \ Relationship Based on
Results of Ariathurai and Arulanandan (1978)
(after Mehta, 1981) 83
3.21 Relative Suspended Sediment Concentration Versus
Time for a Stratified Bed (after Mehta and
Partheniades, 1979) 83
3.22 Schematic Representation of the Selected
Methodology for the Variation of the Applied
Bed Shear Stress During Bed Preparation and
Resuspension Tests (after Mehta et al . ,
1982a) 86
3.23 Variation of Suspension Concentration with Time
for Tj^ = 48 Hours (after Dixit, 1982) 87
3.24 C(T^') Versus t^ for Three Values of Tj^,,
Using Kaolinite in Salt Water (after Mehta
et_al_., 1982a) 88
xi
FIGURE PAGE
3.25 Normalized Rate of Erosion, e^/£ Versus
Normalized Excess Shear Stress, (zi^) )/
T (zu). Using Kaolinite in Tap Water (after
Mehta et_ al_., 1982a) 90
3.25 Normalized Rate of Erosion, Versus
Normalized Excess Shear Stress, ^'^i~'^A^h'^^ ^
^(^U^), Using Kaolinite in Salt Water (after
Mehta et al_. , 1982a) 91
3.27 Critical Shear Stress Versus SAR for
Montmorillonitic Soil (after Alizadeh,
1974) 94
3.28 Resuspension Rate Versus Normalized Excess Shear
Stress 97
3.29 Slope, a , Versus Depth Below Bed Surface, z^,
as a Function of Salinity 98
3.30 Ordinate Intercept, e^. Versus Depth Below Bed
Surface, z^, as a Function of Salinity 98
3.31 The Internal Circulation Driven by the River
Discharge in a Partially Stratified Estuary,
(a) A Transverse Section, (b) A Vertical
Section (after Fischer et_ al_., 1979) 110
3.32 Illustration of Windinduced Circulation (after
Fischer, 1972) 110
3.33 Ratio C/Cg Versus Time t for Kaolinite in
Distilled Water (after Mehta and Partheniades,
1975) 120
3.34 Ratio Cg /C^ Versus Bed Shear Stress v (after
Mehta and Partheniades, 1975) 120
3.35 Relative Steady State Concentration C^g in
Percent Against Bed Shear Stress Parameter \i
(after Mehta and Partheniades, 1975) 122
3.36 C in Percent Versus t/t5Q for Kaolinite in
Distilled Water (after Mehta and Partheniades,
1975) 124
3.37 Log t^q Versus for Kaolinite in Distilled
Water (after Mehta and Partheniades, 1975) 126
3.38 cj^ Versus for Kaolinite in Distilled
Water (after Mehta and Partheniades, 1975) 126
xi i
FIGURE PAGE
3.39 Settling Velocity, W^, Versus Suspended Sediment
Concentration, C, for San Francisco Bay Mud
(after Krone, 1962) 127
3.40 Settling Velocity, W^, Versus Suspended Sediment
Concentration, C, for Yangtze River Estuary Mud
(after Huang etal_., 1980) 129
3.41 Versus C for Severn Estuary Mud
(after Thorn, 1981) 129
3.42 Effect of Size and Settling Velocity of Elementary
Particles on the Coagulation Factor of Natural
Muds (after Bellessort, 1973) 132
3.43 Effect of Salinity on Settling Velocity of San
Francisco Bay Mud (after Krone, 1962) 134
3.44 Effect of Salinity on Settling Velocity of
Avonmouth Mud (after Owen, 1970) 136
3.45 Effect of Salinity and Suspension Concentration
on Settling Velocity of Avonmouth Mud (after
Owen, 1970) 137
3.46 Ratio C/Cg Versus Time as a Function of the Bed
Shear, ti^, for Lake Francis Sediment with
3=5 ppt 139
3.47 Ratio C /C^ Versus for Deposition Tests
with LaRe Francis Sediment 141
3.48 Apparent Settling Velocity Description in Domains
Defined by Suspended Sediment Concentration and
Bed Shear Stress 141
3.49 Effect of Salinity and Bed Shear Stress on
Settling Velocity of Lake Francis Sediment 146
3.50 Settling Velocity Versus Suspension Concentration
for Deposition Test with Lake Francis Sediment. . . ISO
3.51 Variation of 0*^ with Salinity and x^ 150
3.52 Variation of Mean Bed Density with Consolidation
Time (after Dixit, 1982) 157
3.53 Variation of p/p^ ^^^h Consolidation Time (after
Dixit, 1982) 157
xi ii
FIGURE
PAGE
3.54 z^/W Versus p/p for Avonmouth, Brisbane,
Grangemouth and Belewan Muds (after Dixit,
1982) 158
3.55 Zu/H Versus p/p for Consolidation Times (a) Less
TFian 48 Hours and (b) Greater Than 48 Hours
(after Dixit, 1982) 159
3.56 Normalized Bed Density Profiles for Thames Mud for
Two Different Consolidation Times 160
3.57 Normalized Bed Density Profiles for Avonmouth Mud
as a Function of Salinity 160
3.58 Normalized Bed Density Profiles for Avonmouth Mud
for Different Bed Thicknesses 161
3.59 Variation of ■^r^zu) with Zj^ for Various
Consolidation Periods (after Dixit, 1982) 165
3.60 Correlation of Bed Shear Strength with Bed
Density (after Owen, 1970) 167
3.61 Variation of p(Z^) with Incorporated in
Consolidation Algorithm 172
3.62 Bed Schematization Used in Bed Formation 
Consolidation Algorithms 176
4.1 Global and Local Coordinates 190
5.1 Downstream View of Recirculating Flume. Width
Reducing Device is Shown on Right Side of
Flume 205
5.2 Schematic Diagram of Recirculating Flume (after
Dixit, 1982). 206
5.3 Kent MiniFlow Current Meter 208
5.4 Calibration of Kent MiniFlow Current Meter .... 208
5.5 Instrumentation Cart and Setup of Kent MiniFlow
Meter and Two Point Gages 210
5.6 Electric Point Gage and Tube of Water Surface
Elevation Measuring Device 211
5.7 Setup of Water Surface Elevation Measuring
Device (after Wang, 1983) 211
xi V
FIGURE PAGE
5.8 (a) Apparatus I for Obtaining Sediment Core;
(b) Apparatus II for Sectioning a Frozen Sediment
Core (after Parchure, 1980) 213
5.9 Water Sampling Device 215
5.10 Grain Size Distribution of Kaolinite Used for
the Experiments 215
5.11 Shear Stress History for Experiments in the
Recirculating Flume 219
5.12 Measurement Stations in the Recirculating
Flume 221
5.13 Measured and Predicted Suspended Sediment
Concentrations for Test No. 1 223
5.14 Measured and Predicted Suspended Sediment
Concentrations for Test No. 2 224
5.15 Measured and Predicted Suspended Sediment
Concentrations for Test No. 3 225
5.16 Measured Bed Density Profiles for Experiments in
the Recirculating Flume 228
5.17 Rotating Annular Flume 229
5.18 VelocityTime Record, and the Measured and
Predicted Suspended Sediment Concentrations in
the Tidal Cycle Experiment 232
5.19 Finite Element Grid of Recirculating Flume;
Distorted Sketch  Width: Length =4.1:1.0 234
5.20 Schematic Representation of Set up for Experiment
in the 100 m Flume (after Dixit et al . ,
1982) 237
5.21 Measured and Predicted Deposit Thickness Along
100 m Flume .... 239
5.22 Aerial View of Camachee Cove Marina 247
5.23 Bathymetry of Camachee Cove Marina Obtained in
September, 1982 (after Srivastava, 1983). ..... 248
5.24 Finite Element Grid of Camachee Cove Marina .... 250
5.25 Predicted Sedimentation Contours for Marina
Basin 251
XV
t^IGURE PAGE
5.26 Plan View of 10 km Hypothetical Canal 253
5.27 Predicted Suspended Sediment Concentrationtime
Record for Element No. 4 in Hypothetical
Canal 255
5.28 Predicted Suspended Sediment Concentrationtime
Record for Element No. 5 in Hypothetical
Canal 256
xvi
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
PREDICTION OF COHESIVE SEDIMENT MOVEMENT
IN ESTUARIAL WATERS
By
Earl Joseph Hayter
December 1983
Chairman' Dr. B.A. Christensen
CoChairman : Dr. A.J. Mehta
Major Department: Civil Engineering
Fine sediment related problems in estuaries include shoaling in
navigable waterways and water pollution. A twodimensional (horizontal)
fine, cohesive sediment transport model using the finite element method
has been developed to predict the temporal and spatial variations of the
depthaveraged suspended sediment concentrations in estuarial waters.
The advectiondi spersion equation with appropriate source/sink terms is
solved by the Galerkin weighted residual method for the suspension
concentration at each node. Contemporary laboratory and field evidence
has been used to develop algorithms which describe the processes of
erosion, dispersion, settling, deposition, bed formation and bed
consolidation.
XV i 1
The model yields stable and converging solutions. A useful feature
of the model is its ability to predict the influence of salinity on the
rate of fine suspended sediment movement. Verification was carried out
against results from a series of erosiondeposition experiments in the
laboratory using kaolinite and a natural mud as the sediments. The
model was applied under prototype conditions to simulate sedimentation
in a marina basin and suspended sediment transport in a hypothetical
canal in which both erosion and deposition occurred.
XV i i i
CHAPTER I
INTRODUCTION
1.1. Estuarial Cohesive Sediment Dynamics
Cohesive sediments in estuaries are comprised largely of
terrigenous claysized particles. The remainder may include fine silts,
biogenic detritus, algae, organic matter, waste materials and sometimes
small quantities of very fine sand. Although in water with a very low
salinity (less than about 1 part per thousand) the elementary sediment
particles are usually found in a dispersed or "nonsalt flocculated"
state, small amounts of salts are sufficient to repress the
electrochemical surface repulsive forces between the elementary
particles, with the result that the particles coagulate to form much
larger aggregates. Each aggregate may contain thousands or even
millions of elementary particles. The transport properties of
aggregates are affected by the hydrodynamic conditions and by the
chemical composition of the suspending fluid. Most estuaries contain
abundant quantities of cohesive sediments which usually occur in the
coagulated form in various degrees of aggregation. Therefore, an
understanding of the transport properties of these sediments in
estuaries requires a knowledge of the manner in which the aggregates are
transported in these waters.
Cohesive sediment transport in estuaries is a complex process
involving a strong coupling between tides, baroclinic circulation and
the coagulated sediment. For an extensive description of this process,
1
2
the reader is referred to Postma (1967), Parthem'ades (1971), Barnes and
Green (1971), Krone (1972), Kirby and Parker (1977), and Kranck
(1980). In Fig. 1.1, a schematic description is given. The case
considered is one in which the estuary is stratified, and a stationary
saline wedge is formed as shown. Various phases of suspended fine
sediment transport are shown, assuming a ti dailyaveraged situation. In
the case of a partially mixed estuary, the description will be modified,
but since relatively steep vertical density gradients are usually
present even in this case, the sediment transport processes will
generally remain qualitatively similar as depicted in Fig. 1.1.
As indicated in Fig. 1.1, riverborne sediments from upstream fresh
water sources arrive in the mixing zone of the estuary. The
comparatively high degree of turbulence, the associated shearing rates
and the increasingly saline waters will cause aggregates to form and
grow in size as a result of frequent interparticle collisions and
increased cohesion. The large aggregates will settle to the lower
portion of the water column because of their high settling velocities.
Results based on laboratory experiments show that aggregate settling
velocities can be up to four orders of magnitude larger than the
settling velocities of the elementary particles (Bellessort, 1973).
Some of the sediment will deposit and some will be carried upstream near
the bottom until periods close to slack water when the bed shear
stresses decrease sufficiently to permit deposition. The sediment will
start to undergo selfweight consolidation. The depth to which the new
deposit scours when the currents increase after slack will depend on the
bed shear stresses imposed by the flow and the shear strength of the
deposit. Net deposition, i.e. shoaling, will occur when the bed shear
3
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during flood, as well as during ebb, is insufficient to resuspend all of
the material deposited during preceding slack periods. Some of the
sediment that is resuspended will be reentrained throughout most of the
length of the mixing zone to levels above the salt waterfresh water
interface and will be transported downstream to form larger aggregates
once again, and these will settle as before. At the seaward end some
material may be transported out of the system, a portion or all of which
could ultimately return with the net upstream bottom current.
In the mixing zone of a typical estuary the sediment transport
rates often are an order of magnitude greater than the rate of inflow of
"new" sediment derived from upland or oceanic sources. The estuarial
sedimentary regime is characterized by several periodic (or quasi 
periodic) macrotimescales, the most important of which are the tidal
period (diurnal, semidiurnal, or mixed) and onehalf the lunar month
(springneapspring cycle). The first is of course the most important
since it is the fundamental period which characterizes the basic mode of
sediment transport in an estuary. The second is important from the
point of view of determining net shoaling rates in many cases of
engineering interest.
1.2. Sediment Related Problems in Estuaries
Estuaries are often centers of population and industry, and as such
are used as commerce routes to the sea, convenient dump sites for waste
products as well as for man's recreational enjoyment. They also serve
as the sink for sediment and pollutants transported by rivers from
inland sources. As man's activity in and hence dependence upon
estuaries has increased with the growth of population and commerce in
5
these areas, the necessity of proper estuan'al management becomes very
ostensible. Included in estuarial management is the maintenance of
navigable waterways and water pollution control, both of which are
affected to varying degrees by the load of suspended and deposited
sediment. These two tasks are examined next.
Under low flow velocities, sometimes coupled with turbulent
conditions which favor the formation of large aggregates, cohesive
sediments have a tendency to redeposit in areas such as dredged cuts and
navigation channels, in basins such as harbors and marinas, and behind
pilings placed in water (Einstein and Krone, 1962; Ariathurai and F^ehta,
1983). In addition, as noted previously, the mixing zone between upland
fresh water and sea water in estuaries is a favorable site for bottom
sediment accumulation. Inasmuch as estuaries are often used as
transportation routes, it is desirable to be able to accurately estimate
the amount of dredging required to maintain navigable depths in these
water bodies, and also to predict the effect of new estuarial
development projects such as the construction of a port facility or
dredging of additional navigation channels.
Cohesive sediments may influence water quality by affecting aquatic
life and by providing a large assimilative capacity as well as
transporting mechanism for dissolved and suspended pollutants.
Turbidity caused by suspended sediment particles restricts the
penetration of light, and therefore reduces the depth of the photic
zone. This in turn may result in a decrease in the production of
phytoplankton and other algae which leads to a reduction in the amount
of food available for fish. Deposited sediments can damage spawning
areas for fish and eliminate invertebrate (e.g. oysters) populations.
6
The bulk of the pollution load in a water body is quite often
transported sorbed to cohesive sediments rather than in the nonsorbed
state (Preston etal_., 1972; Kirby and Parker, 1973). Therefore, the
importance of considering the movement of cohesive sediments in
predicting the fate of pollutants (e.g. pesticides, radioisotopes, and
toxic elements such as lead, mercury, cadmium, nickel and arsenic)
introduced in an estuary cannot be overemphasized. The properties of
cohesive sediments, and in particular clays, which cause the sorption of
pollutants are the large surface area to volume ratio, the net negative
electrical charges on their surfaces and their cation exchange
capacity. These properties are discussed in Chapter II.
In an investigation of the bottom sediments from several coastal
marinas in Florida, two interesting observations were made (Weckmann,
1979; Bauer, 1981). First, when comparing sediment particle size inside
the basin with that obtained immediately outside in the main body of
water, it was found that in the majority of the marinas investigated,
the sediment inside was measurably finer than that outside. Second, a
similar comparison in terms of heavy metal (e.g. Cu, Pb, Ni, Cd and Zn)
content within the basin and without indicated measurably higher
concentrations inside the basin. These two observations, when
correlated, exemplify the role of cohesive sediments in accumulating
pollutant levels in estuarial depositional environments such as marina
basins. This assimilation of pollutants and storage in bottom sediments
may prove to be an acceptable means of waste disposal, providing the
contaminated bottom sediment is not resuspended and the pollutant
desorbed. However, sometimes even a relatively small change in the
chemical composition of the water may cause desorption of pollutants
from the sediment particles.
7
1.3. Approach to the Problems
Prediction of the fate of sorbed pollutants or the frequency and
quantity of dredging required to maintain navigable depths in a channel
or harbor can be accomplished by modeling the movement of cohesive
sediments in the water body of concern. It becomes necessary to
simulate the various transport processes, i.e. erosion, advective and
dispersive transport, aggregation, deposition and consolidation, and the
physical factors, e.g. movement of water and dissolved salt, that govern
these processes. The movement of suspended sediment, water and salt are
highly interrelated, as is evident upon examination of Fig. 1.2 which
defines possible interactions between these constituents in an estuary.
Physical and mathematical models or combinations (hybrid approach)
of these two types are the types of models available for use in
predicting cohesive sediment movement in a water body. Physical scale
models have only been partially successful due to lack of an appropriate
model sediment as well as due to poor model reproduction of estuarial
mixing processes and internal shear stresses (Owen, 1977). Mathematical
models, however, have been generally more successful in reproducing,
with some degree of accuracy, the movement of cohesive sediments in
estuarial waters. The modeling philosophy is delineated below.
To mathematically model the motion of the three main constituents
in an estuarial environment the threedimensional forms of the
conservation of momentum and mass equations for the water and the
conservation of mass equations for the dissolved salt, suspended
sediment and pollutant, if present, must be solved numerically. However,
due to the current high cost of solving such threedimensional, coupled,
partial differential equations, only a few threedimensional models
8
Gradual
Large
Motion of
dissolved
sal t
Rapid
Large
Large
Bulk flow
tidal
propagation
Rapid
Large
Rapid
aMedi um
Large
Rapid
Velocity field.
Internal water
ci rculation.
Bed shear
Rapid
Medium
Rapid
Large
Rapid
Large
Medium
Gradual
Medi um
Large
Rapid
Very
Gradual
Smal 1
Motion of
suspended
mud
Larae
Rapid
Large
Rapid
Large
Rapid
Medi um
Coagulation'
and
settling /Medium
Gradual
Medium
Fig. 1.2. Interactions of Tidal and Estuarial Sediment Transport
Processes (after Owen, 1977).
9
exist (L1u and Leendertse, 1978). The common procedure has been to
spatially integrate these equations, laterally and/or vertically, in
order to reduce them to their two or onedimensional forms.
The horizontal length scales relative to the transport of cohesive
sediments typically are one to three orders of magnitude greater than
the vertical length scales in most estuaries. As a consequence, and
because horizontal transport distances are usually of primary interest
in ascertaining the magnitude of sedimentation or the fate of sorbed
pollutants, it is in most cases not unreasonable to use vertically
integrated transport equations for modeling purposes. However, even
using the twodimensional forms of the governing equations, some eight
to ten coupled equations must be solved to completely model the depth
averaged motion of water, sediment and salt. As a result, the modeling
of water and salt movement is commonly performed separately from the
sediment transport modeling. For example, a twodimensional
hydrodynamic model, which solves the coupled momentum and (water and
dissolved salt) continuity equations, is used to model the movement of
water and salt. Then a twodimensional cohesive sediment transport
model would be used to predict the motion of sediment using the results
from the hydrodynamic model.
1.4. Scope of Investigation
Mathematical descriptions of the physical processes prevalent in a
binary fluidsediment system (such as the flux of sediment to and from
the bed, and the dispersion, aggregation and settling of suspended
aggregates in a turbulent flow field) that are incorporated in existing
suspended cohesive sediment transport models, many of which are
10
described in Chapter IV, are limited as they use empirical evidence
based on limited studies conducted prior to the early 1970's. Since
that time, a considerable amount of experimental research has been
conducted, partly at the University of Florida, on the various aspects
of cohesive sediment transport mechanics. Utilization of contemporary
laboratory experimental and field evidence to develop new algorithms
which describe the transport processes of erosion, dispersion, settling,
deposition and bed consolidation would result in a model whose
predictive capability is measurably improved over that of existing
model s.
The intent of this investigation therefore was twofold: 1) to
develop a twodimensional depthaveraged cohesive sediment transport
model using stateoftheart information on estuarial cohesive sediment
processes, and 2) to verify this model with results from
erosion/deposition experiments performed in a 18 m long recirculating
flume and in an annular rotating channel, and a deposition experiment
performed in a 100 m long flume. The sediment process information
incorporated in the model is based on a detailed analysis and
interpretation of available laboratory and field data. The results of
this research is presented in the following format.
In Chapter II, a description of the composition, structure and
physicochemical properties of cohesive sediments is given. This is
followed by discussions on estuarial dynamics and sedimentation
processes.
Chapter III begins with a description of the governing equations
for sediment transport in an estuarial environment. Then, previous
investigations on erosion, dispersive transport, deposition, and the
11
structure and consolidation of cohesive sediment beds are described,
followed by detailed descriptions of the sediment transport algorithms
developed in this investigation.
In Chapter IV, the new cohesive sediment transport model is
described in detail. Included in this chapter is a review of previous
models, and descriptions of the finite element solution routine and the
stability and convergence characteristics of the new model.
In Chapter V, the model is verified using the results from: 1)
experiments performed by the investigator in a 18 m long recirculating
flume at the University of Florida, and 2) a deposition experiment
performed in a 100 m long flume at the U.S. Army Corps of Engineers
VJaterways Experiment Station in Vicksburg, Mississippi. The model is
then used to simulate the sedimentation in a coastal marina. Lastly,
limitations of the modeling approach, as well as possible model
applications to water quality and sedimentation management problems are
di scussed.
In Chapter VI, conclusions from this study and recommendations for
future research are presented.
In Appendix A, the twodimensional depthaveraged form of the
advectiondispersion equation is derived. In Appendix B, the element
coefficient and load matrices are given. In Appendix C, a description
of the computer program, including a user's manual is given. Lastly, in
Appendix D, the field data collection and laboratory analysis programs
required to develop the data base for the model are described.
CHAPTER II
BACKGROUND MATERIAL
2.1. Introductory Note
Understanding the movement of cohesive sediments 1n an estuarlal
environment requires knowledge pertaining to the physicochemical
properties of cohesive sediments, estuarial hydrodynamics and sediment
transport processes. Each of these topics is briefly discussed below.
2.2. Description and Properties of Cohesive Sediments
2.2.1. Composition
As noted in Chapter I, cohesive sediments consist primarily of
claysized material. Such material consists of clay and nonclay
mineral components and organic material (Grim, 1968). Clay particles
are generally less than 2 microns {\m) in size. As a result they are
termed colloids, and in water possess the properties of plasticity,
thixotropy and adsorption (van Olphen, 1963). Grim (1968) states that
"the term clay implies a natural, earthy, finegrained material", and
that clays are "composed essentially of silica, alumina, and water,
frequently with appreciable quantities of iron, alkalies, and alkaline
earths." The most abundant types of clay minerals are kaolinite,
montmoril lonite, illite, chlorite, vermiculite, and halloysite.
Nonclay minerals consist of, among others, quartz, carbonates,
feldspar, and mica (Grim, 1968). This component of clay material is
generally larger than 2 i^m in size, though this is not so in all clay
12
13
mateHals. It H currently not possible to quantitatively determine,
with a high degree of accuracy, the amount of nonclay minerals present
in a clay mateHal. Grim (1958) points out the consequence of this
1t«ttatfon by stating that "the absence of accurate quantitative methods
for determining the nonclay mineral components of clay materials
frequently makes ft impossible m obtain exact data on the chemical
composition of the clay minerals themselves in such materials."
The organic material usually present in clay materials may sxfst as
discrete particles of matter (e.g. wood), as sorbed organic molecoles on
the surface of the clay ptrtfdes. or' inserted between clay layers
(Gnm. 1968). The percentage by weight of organic matter in a clay
material may be determined through use of a standard analytical
procedure such as the WalkleyRlack test (Allison. 1965).
Additional possible components of clay materials are watersoluble
salts, and sorbed exchangeable ions and pollutants. Watersoluble salts
include chlorides, sulfates, alkaline earths, carbonates of alkalies,
aluminum and iron. The most common exchangeable cations and anions in
clay materials are. respectively. Ca^^ Mg2^ n\ K*. mH,"^ and Cr, SO^".
NO3 and P0,3 (Grim. 1968). The presence of pollutants sorbed to the
surface of clay particles was discussed in Chapter 1.
2.2. g. Origin
In nature clay materials are produced by hydrothermal action and
weathering of rocks. Fact0rs which Influence the type of clay minerals
formed by these two processes include the composition of the parent
rock, m Climate, topography, the abundance and kind of vegetition,
time, and pH of the ground water, the presence and kind of alkalies and
14
alkaline earths, the intensity of the hydrothermal alteration, and the
permeability and porosity of the host rock. Grim (1968) describes the
origin and occurrence of clay materials in detail.
2.2.3. Structure
Clay minerals are primarily hydrous aluminum silicates with
magnesium or iron occupying all or part of the aluminum positions in
some clays, and with alkalies (e.g. sodium, potassium) or alkaline
earths (e.g. calcium, magnesium) also present in others (Grim, 1968).
Most clays are composed of one of two atomic structural units, or
combinations of the two basic units. These are the silica tetrahedron
and the aluminum hydroxide octahedral unit. The former consists of a
central silicon atom surrounded by four oxygen atoms or hydroxy Is
arranged in a tetrahedral configuration. The tetrahedrons are bonded
together in a hexagonal network in such a way that a sheet structure of
composition Si40g(0H)4 is formed. The structure of each sheet is such
that the oxygens and/or hydroxy Is forming the bases of the tetrahedral
units are in the same plane, and the tips of all the units point in the
same direction (Grim, 1968).
The octahedral aluminum hydroxide unit consists of "two sheets of
closely packed oxygens or hydroxyls in which aluminum, iron, or
magnesium atoms are embedded in octahedral coordination, so that they
are equidistant from six oxygens or hydroxyls" (Grim, 1968). If
aluminum atoms are present, only two out of every three central
positions will be filled so that the structure's electrical charge will
be balanced. However, if magnesium atoms are present, all central
positions are occupied. With aluminum, the octahedral is known as the
15
gibbsite structure, which has the formula Alj(OH)g. The brucite
structure is that formed with magnesium, and has the formula Mg3(0H)g
(Grim, 1968).
The different clay minerals are formed by stacking of the sheet
structures to form layers, and substitution of different ions for the
aluminum in the octahedral unit. Grim (1968) gives an excellent
description of the structural arrangement of the most common types of
clay minerals.
Ions of one kind are sometimes substituted by ions of another kind,
with the same or different valence. This process does not necessarily
involve replacement. The tetrahedral and octahedral cation
distributions develop during initial formation of the mineral, and not
by later substitution (Mitchell, 1976). Substitution in all the clay
materials, except for kaolinite, gives clay particles a negative
electric charge which is of great significance in coagulation of clays
and in absorption of pollutants. Another cause of net particle charge
is the preferential sorption of peptizing ions on the surface of the
particle (van Olphen, 1963).
2.2.4. Interparticle Forces
For particles in the colloidal size range, surface physicochemical
forces exert a distinct influence on the behavior of the particles due
to the large specific area, i.e. ratio of surface area to volume. As
stated previously, most clay particles fall within the colloidal range
in terms of both their size (2 lim or less) and the controlling influence
of surface forces on the behavior. In fact, the average surface force
6
on one clay particle is approximately 10 times greater than the
gravitational force (Partheniades, 1962),
16
The relationship between clay particles and water molecules is
governed by the interparticle electrochemical forces. The different
configurations and groupings as well as electric charges of clay
particles affect their association with water molecules (Grimshaw,
1971). Water molecules possess a permanent electrical imbalance or
dipole moment which results from the molecular arrangement of the oxygen
and hydrogen atoms. According to Grim (1962), the electrostatic field
emanating from the surface of a clay particle orients the polar water
molecules in the pores separating adjacent particles.
Interparticle forces consist of both attractive and repulsive
forces. The attractive forces present are the Londonvan der Waals, and
are due to the nearly instantaneous fluctuation of the dipoles which
result from the electrostatic attraction of the nucleus of one atom for
the electron cloud of a neighboring atom (Grimshaw, 1971). These
electrical attractive forces are weak, and are only significant when
interacting atoms are very close together. However, they are strong
enough to cause structural buildup as they are additive between pairs
of atoms. Thus, the total attractive force between two clay particles
is equal to the sum of the attraction between all the atoms comprising
both particles. This additive effect results in a larger attractive
force and to a smaller decrease in this force with increasing particle
separation. Figure 2.1 shows qualitatively the relationship between the
attractive energy V^^ of one particle for another and the particle
separation distance. The attractive energy is inversely proportional to
the sixth power of the separation distance for two atoms and to the
second power for two spherical particles. The magnitude of decreases
with increasing temperature and is dependent upon the geometry and the
17
Double Layer Repulsion at
Three Different Electrolyte
Concentrations
\ \ Internnediate
VliqtNN^ Particle Seoar a t ion
van der wools
Attraction
Fig. 2.1. Repulsive and Attractive Energy as a Function of Particle
Separation at Three Electrolyte Concentrations (after
van Olphen, 1953) .
V
Net Interaction
Energy
Particle Seoaration
mm
Fig. 2.2. Net Interaction Energy as a Function of Particle Separation
at High Electrolyte Concentration (after van Olphen, 1953).
18
size of adjacent clay particles. The attractive energy has been found
to be only slightly dependent upon the salt concentration (i.e.
salinity) of the medium (van Olphen, 1963).
The repulsive forces of clay materials are due to the negatively
charged particle forces. The repulsion potential increases in an
exponential fashion with decreasing particle separation. The magnitude
of these forces is dependent upon the salinity, decreasing with
increasing salinity as shown in Fig. 2.1, where is the repulsive
energy. This dependence of Vj, on the salinity can best be explained
using the concept of the electrical double layer and the surrounding
diffuse layer, van Olphen (1963) states that the double layer is
composed of the net electrical charge of the elementary clay particle
and an equal quantity of ionic charge of opposite sign located in the
medium near the particle surface. Thus, the net electrical charge is
balanced in the surrounding medium. The ions of opposite charge are
called the counterions, i.e. cations. The counterion concentration
increases with decreasing distance from the particle surface. This
layer of counterions is referred to as the diffuse layer. A clay
particle and the associated double layer is referred to as a clay
micelle (Partheniades, 1971). When the salinity is increased, the
diffuse layer is compressed toward the particle surface (van Olphen,
1963). The higher the salinity, and as well the higher the valence of
the cations which compose the diffuse layer, the more this layer is
compressed and the greater the repulsive force is decreased.
With a high salinity, corresponding to a value approximately that
of seawater (35 ppt), the attractive forces become predominant at all
but extremely small particle separation distances. The interaction
19
potential, determined by summing and V^, reflects this dominance, and
shows the highest attractive potential (primary minimum) at separation
distances on the order of 1 nm (10~\) (Parker, 1980). At distances
less than this the short range repulsive forces are predominant (van
Olphen, 1963). Figure 2.2 shows this net interaction potential as a
function of particle separation for high salinity. Thus, two clay
particles will adhere when they reach the separation distance at which
the primary minimum occurs. Cohesion or particle destabi 1 ization occurs
at a maximum rate due to the presence of attractive forces even at
relatively great distances.
For medium and low salinities, on the order of 1015 ppt and 12
ppt respectively (Parker, 1980), repulsive forces become predominant at
separation distances of approximately 10 nm where a local repulsive
potential maximum occurs (Figs. 2.3 and 2.4). At distances closer than
this, these interaction potentials are similar to that for high
salinity. As indicated by these figures, the destabi 1 ization of two or
more particles would be expected to decrease for decreasing salinities
as a result of net repulsive forces existing at increasingly larger
distances (van Olphen, 1963).
2.2.5. Cation Exchange Capacity
The cation exchange capacity (CEC) is an important property of
clays by which they sorb certain cations and anions in exchange for
those already present and retain them in an exchangeable state. The CEC
of different clays varies from 315 milliequivalents per 100 grams of
material (meg/100 gm) for kaolinite to 100150 meg/100 gm for
vermiculite. Higher CEC values indicate greater capacity to absorb
20
Fig. 2.3. Net Interaction Energy as a Function of Particle Separation
at Intermediate Electrolyte Concentration (after van Olphen,
1953).
T min
Fig. 2.4. Net Interaction Energy as a Function of Particle Separation
at Low Electrolyte Concentration (after van Olphen, 1963).
21
other cations. The negative surface charge caused by isomorphous
substitution is neutralized by sorbed cations located on the surfaces
and edges of a clay particle. These cations remain in an exchangeable
position and may in turn be replaced by other cations.
The following factors are the causes of cation exchange: 1)
substitution within the lattice structure results in unbalanced
electrical charges in the structural units of some clays, and 2) broken
bonds around the edges of the tetrahedral octahedral units give rise to
unsatisfied charges. In both cases the unbalanced charges are balanced
by the sorbed cations. The number of broken bonds and hence the CEC
increases with decreasing particle size.
The ability to replace exchangeable cations depends on the
concentration of the replacing cation, the number of available exchange
positions, the nature of the anions and cations in the replacing
solution. Increased concentration of the replacing cation results in
greater cation exchange. The release of an ion depends upon the nature
of the ion itself, upon the nature of the other ions filling the
remaining exchange positions, and upon the number of unfilled exchange
sites. The higher the valence of a cation, the greater is its replacing
power and the more difficult it is to displace when sorbed on a clay.
Some of the predominantly occurring cations in sediments are sodium,
potassium, calcium, aluminum, lead, copper, mercury, chromium, cadmium
and zinc.
2.2.6. Coagulation
Coagulation of suspended cohesive sediments depends upon
interparticle collision and interparticle cohesion after collision.
22
Cohesion and collision, discussed in detail by among others Kruyt
(1952), Einstein and Krone (1962), Krone (1962), Partheniades (1964),
O'Melia (1972), and Hunt (1980) are reviewed here.
The collision frequency, I, for suspended sediment particles of
effective diameters d^ and d  is given by (Hunt, 1980):
I = P(d.,d ) dN dN. (2.1)
'J 'J
where P(d^,dj) = collision function determined by the collision
mechanism (discussed below), which has units of fluid volume per unit
time, and dN^ = number of particles with sizes between d^ and d^+d(d^)
per unit volume of the fluid.
There are three principle mechanisms of interparticle collision in
suspension, and these influence the rate at which elementary sediment
particles coagulate. The first is due to Brownian motion resulting from
thermal motions of molecules of the suspending ambient medium. The
collision function corresponding to this mechanism is given by (Hunt,
1980):
2 kT. (d.+d.)^
Pu(d.,d.) = ^ ' ^ (2.2)
° ^ 3 ^ d.d.
where k = Boltzmann constant, T^ = absolute temperature and \^ = dynamic
viscosity of the fluid. Generally, coagulation rates by this mechanism
are too slow to be significant in estuaries unless the suspended
sediment concentration exceeds 10 g/1 . Aggregates formed by this
mechanism are weak, with a lacelike structure and are easily fractured
by shearing in the flow or are crushed easily when deposited (Krone,
1962).
23
The second mechanism is that due to internal shearing produced by
local velocity gradients in the fluid. Collision will occur if the
paths of the particle centers in the velocity gradient are displaced by
a distance which is less than the sum of their radii (referred to as the
where G is the local shearing rate and R^ j = d^.+dj. Aggregates produced
by this mechanism tend to be spherical, and are relatively dense and
strong because only those bonds that are strong enough to resist the
internal shearing due to local velocity gradients can survive. The
frequency of collision is especially high in an estuarial mixing region
where a large number of suspended particles are found.
The third mechanism, differential sedimentation, results from the
fact that particles of different sizes have different settling
velocities. Thus a larger particle, due to its higher settling
velocity, will collide with smaller, more slowly settling particles
along its path and will have a tendency to "pick up" these particles on
its way down. The collision function is expressed as
collision radius, R^j, between d^ and dj size particles). The collision
function is given as
P3,{d..dj)
^ 3
=  R. .
6
(2.3)
Tig p^p^
w
.)(d.+d.)^d.^d.^!
(2.4)
11 1 J
24
where v = kinematic viscosity of the fluid, = floe density and =
fluid density. This mechanism produces relatively weak aggregates and
contributes to the often observed rapid clarification of estuarial
waters at slack.
All three mechanisms operate in an estuary, with internal shearing
and differential sedimentation generally being predominant in the water
column, excluding perhaps the high density nearbed layer, where
Brownian motion is likely to contribute significantly as a collision
mechanism. Then again, internal shearing is probably more important
than differential sedimentation during times, excluding those near slack
water, when collision and coherence due to differential settling would
be expected to be the main mechanism controlling the rate of
coagulation.
Hunt (1980) compared the values of the three collision functions
(Eqs. 2.2  2.4) for collision of a d^.=l m size particle with varying
sizes, dj, of the colliding particle under the following conditions:
temperature 14°C, shearing rate G = 3 sec"", and (P^'P^^fP^ ~ 0.02. The
comparison is shown in Fig. 2.5 and reveals that each collision
mechanism is dominant over a certain particle size range. In this
example, Brownian motion is the dominant mechanism for particles less
than 1 m, internal shearing is dominant for particles between 1 and 100
m, and differential sedimentation is dominant for particles greater
than 100 \m. Hunt states that the same ordering of the dominant
collision mechanisms with increasing dj would be achieved for collisions
with other d^ sizes. Thus, the collision frequency is controlled not
only by the prevailing flow conditions and local suspension
concentration, but by the size of the colliding particles and/or floes
as well.
25
d; (/xm)
Fig. 2.5. Comparison of the Collision Functions for Brownian, Shear
and Differential Sedimentation Coagulation (after Hunt, 1980).
26
Cohesion or particle destabilization of colloidal particles is
caused by the presence of net attractive electrochemical surface forces
on the particles. The latter condition is promoted by increased
concentration of dissolved ions and/or increased ratio of multivalent to
monovalent ions, both of which serve to depress the double layer around
micelles and thus allow the attractive Londonvan der Waals and
coulombic forces to predominate (Krone, 1963). Since sea salt is a
mixture of salts, with monovalent sodium ions and divalent calcium and
magnesium ions prevalent in natural electrolytes, the effect of these
salts on cohesion is determined by the relative abundances of mainly
these three ions (see Table 3.2), the latter being indicated by the
sodium adsorption ratio (SAR). The SAR is defined as
where the cation concentrations are in milliequivalents per liter
(Arulanandan, 1975). The relationship between the SAR and salinity is
seen in Fig. 2.6. The cation exchange capacity (CEC), salinity and SAR
all serve to determine the net interparticle force and thus the
potential for micelles to become cohesive.
Kandiah (1974) found that the boundary between the dispersed and
coagulated states for the three main clay groups, kaolinite, illite and
montmorillonite, varied with the SAR, total salt concentration and pH of
the solution (see Figs. 2.7a, b, and c). The dashed lines in these
figures represent interpolated boundary curves for a pH range of 7.5 to
27
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29
8.5, which 1s the range found in sea water at all salinities. It is
evident that the boundary between the dispersd and coagulated states for
these three clays are different. Kaolinite becomes cohesive at a
salinity of 0.6 pnt, illite at 1.1 ppt and montmorillonite at 2.4 ppt
(Ariathurai, 1974). Whitehouse et_ al_. (1960) and Edzwald et al_. (1974)
reported that the cohesiveness of these micelles develops quickly at the
given salt concentrations, and that little increase in coagulation
occurs at higher salt concentrations, which implies that the micelles
must have attained the maximum degree of cohesion. The rapid
development of cohesion and the low salinities at which the main clay
types become cohesive indicates that cohesion is primarily affected by
salinity variations near the landward end of an estuary where salinities are
less than about 3 ppt.
The above cohesion mechanism is referred to as salt flocculation.
There is another cohesion mechanism that operates in water between
micelles in the absence of salt, and hence it is termed nonsalt
flocculation. However, in an estuary the conditions are conducive for
destabilization to be caused by salt flocculation (i.e. depression of
the diffuse double layer). Both types of destabilizing mechanisms are
reported in detail by Lambe (1953).
In summary, it is apparent that cohesive sediment transport in
estuaries is strongly influenced by the coagulation behavior of
dispersed sediment particles, which is controlled by the salinity field,
velocity gradients and the concentration of suspended sediments. In
particular, the salinity of the suspending fluid affects the process of
coagulation in two ways: 1) elementary clay particles become cohesive
when the salinity is equal to or greater than 13 ppt, and 2) the
30
presence of high velocity gradients in the estuarial mixing zone
increases the collision frequency betv;een dispersed particles and/or
aggregates.
2.3. Significance of Important Physical Factors in
Estuarial Transport
2.3.1. Estuarial Dynamics
The hydrodynamic regime in an estuary is governed by the
interaction between fresh water flow, astronomical tides, windgenerated
surface waves, surface (i.e. wind) stresses, Coriolis force, the
geometry of the water body and the roughness characteristics of the
sedimentary material composing the bed (Dyer, 1973). Geometry includes
the shape and the bathymetry of the estuary. The geometry and bed
roughness interact with the driving forces  the first five factors  to
control the pattern of water motion (in particular the shear stress and
turbulence structure near the bed), frictional resistance, tidal damping
and the degree of tidal reflections (Ippen, 1966).
The magnitude of the tidal flow relative to the fresh water inflow
governs, to a large extent, the intensity of vertical mixing of the
lower high density layer with the upper less dense layer. There exists
in all estuaries a horizontal, i.e. longitudinal, salinity profile which
decreases from the mouth to the upper reaches of the estuary. Such
profiles have been measured in numerous estuaries worldwide. A few
examples included here are: Cumbarjua canal, Goa, India during the dry
season, i.e. October through June (Fig. 2.8); the San Francisco Bay and
SacramentoSan Joaquin Delta (Fig. 2.9); the Pamlico River Estuary,
North Carolina (Fig. 2.10); and the Yangtze River Estuary, China
31
Fig. 2.8. Monthly Salinity Distributions in the Cumbarjua Canal,
Goa, India; Ebb;  Flood (after Rao et al. , 1975).
32
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DISTANCE DOWNSTREAM (km)
Fig. 2.10. Salinity of the Surface Waters of the Pamlico River Estuary
as a Function of the Distance from the Railroad Bridge at
Washington, D.C. (after Edwald et al_. , 1974).
40
DISTANCE SEAWARD FROM JIANG ZHEN DOJG(km)
Fig. 2.11. Computed Longitudinal Salinity Profile in the Yangtze River
Estuary as a Function of the Distance Downstream from Jiang
Zhen Dong for Two River Discharges (after Huang et , 1980).
34
(Fig. 2. 11). The existence of a longitudinal salinity gradient, or
baroclinic force, implies that there could be a gravity driven upstream
transport of a high density sediment suspension in the lower portion of
the water column (Officer, 1981; Mehta and Hayter, 1981).
Winds affect the hydrodynamic regime and mixing in an estuary by
generating a surface shear stress and waves. The surface stress is
capable of generating a surface current (whose magnitude will be
approximately three percent of the wind speed at 9.1 m elevation
(Hughes, 1956)) and a superelevation of the water surface along a land
boundary located at the downwind end of the estuary (Ippen, 1966). The
latter effect causes a vertical circulation cell, with landward flow at
the surface and a reversed seaward flow along the bottom. This
phenomenon as well increases the degree of vertical mixing.
Along the banks and in shallow areas, surface gravity waves induced
by the wind are capable of eroding bottom sediments. Since a tidal
current of sufficient strength to transport (but not necessarily to
erode the sediment by itself) suspended sediment is generally present,
this material is advected and dispersed both longitudinally with the
main tidal flow and sometimes laterally with secondary currents towards
the deeper sections of the estuary. Wave action and in particular wave
breaking substantially increase the intensity of surficial turbulence
and mixing.
The Coriolis force, caused by the earth's rotation, has both a
radial (horizontal) and a tangential (vertical) component. The latter
is generally negligible as it is linearly proportional to the vertical
component of the flow velocity, which is typically an order of magnitude
smaller than the horizontal velocity components. The magnitude of the
35
radial component depends upon the size of the water body. Most extra
tropical estuaries are relatively large and therefore the effect of this
force on the hydrodynamic regime is measurable. Estuarial hydrodynamics
are described in extensive detail in such texts as Ippen (1966), Barnes
and Green (1971), Dyer (1973), Officer (1976) and Fischer et al . (1979).
2.3.2. Sediment Processes
The sedimentary regime in an estuary is controlled by the
hydrodynamics, the chemical composition of the fluid and the
physicochemical properties of the cohesive sediment. These factors
affect the processes of erosion, advection, dispersion, aggregation,
settling., deposition and consolidation of the deposited bed. These
processes are briefly described below, following a definition of a clay
suspension.
A "solution" of clay in a medium consists of a homogeneous
dispersion of very small kinetic units, i.e. particles (van Olphen,
1963). When the Stokes diameter of the clay solution is less than 2 \m,
the clay dispersion is usually referred to as a sol. The Stokes
diameter of an arbitrarily shaped particle is determined by equating the
particle's settling velocity with Stokes law for spherical particles and
solving for the "equivalent spherical diameter" (Stokes diameter). When
this diameter is greater than 2 ^m, the dispersion is called a
suspension. However, through use, the term suspension has become
synonymous with dispersion, and thus a clay suspension refers to both
sol and suspension.
Erosion of cohesive soils occurs whenever the shear stress induced
by fluid flow over the bed is great enough to break the electrochemical
36
interparticle bonds (Partheniades, 1965; Paaswell, 1973). When this
happens, erosion takes place by the removal of individual sediment
particles and/or aggregates. This type of erosion is time dependent and
is defined as surface erosion or resuspension. In contrast, another
type of erosion occurs more or less instantaneously by the removal or
entrainment of relatively large pieces of soil. This process is
referred to as mass erosion or redispersion and occurs when the flow
induced shear stresses on the bed exceed the soil bulk strength along
some deepseated plane.
Once eroded from the bed, cohesive sediment is transported entirely
as suspended load (not as bed load) by the estuarial flow. Such
transport is the result of three processes: 1) advection  the sediment
is assumed to be transported at the speed of the local mean flow,
2) turbulent diffusion  driven by spatial suspended sediment
concentration gradients, the material is diffused laterally across the
width of the flow channel, vertically over the depth of flow and
longitudinally in the direction of the transport, and 3) longitudinal
dispersion  the suspended sediment is as well dispersed in the flow
direction by spatial velocity gradients (Ippen, 1966).
In fresh water, most clay particles are in a stabilized or
dispersed state because the repulsive electrochemical surface forces
between the particles prevent them from adhering to one another upon
collision. In the increasingly saline conditions encountered moving
seaward in estuaries, the repulsive forces are suppressed and clay
particles coagulate to form floes. A systematic "build up" of floes as
occurs in estuaries is defined as aggregation. An aggregate is
considered to be the structural unit formed by the joining of floes.
37
The rate and degree of aggregation are two important factors which
govern the transport of cohesive sediments in estuaries. Factors,
besides the water chemistry and the magnitude of the surface forces,
known to govern coagulation and aggregation include sediment size
grading, mineralogical composition, particle density, organic content
and the suspended concentration (i.e. availability) of the sedimentary
material, the water temperature, height through which the floes have
settled, and the turbulence intensity (represented by the shearing rate
G) of the suspending flow (Owen, 1971).
Given the mechanisms which influence the rate of aggregation in an
estuary, the order of aggregation, which characterizes the packing
arrangement, density and shear strength of aggregates, is determined by:
1) sediment type, 2) fluid composition, 3) local shear field, and 4)
concentration of particles or floes available for aggregation. With
regard to the second factor. Krone (1962; 1978) found that the structure
of aggregates is dependent on the salinity for salinities less than
about 10 ppt.
Primary or 0order floes are highly packed arrangements of
elementary particles, with each floe consisting of perhaps as many as a
million particles. Typical values of the void ratio (volume of pore
water divided by volume of solids) have been estimated to be on the
order of 1.2. This is equivalent to a porosity of 0.55, which is a more
"open" structure than commonly occurs in eohesionless sediments (Krone,
1963). Continued aggregation under favorable shear gradients can result
in the formation of first or higher order aggregates composed of loosely
packed arrays of 0order floes. Each succeeding order consists of
aggregates of lower density and lower shear strength. Experimental
38
observations (Krone, 1963; 1978) tend to indicate the following
approximate relationship between the aggregate shear strength, t^, and
aggregate density, p^^, for many (although not all) sediments
% = (P,l)^ (2.6)
where a and p are coefficients which must be determined experimentally
for each sediment. Inasmuch as the shear field in an estuary exhibits
significant spatial and temporal variations, a range of aggregates of
different shear strengths and densities are formed, with the highest
order determined by the prevailing shearing rate, G = du/dz, provided
that: 1) the sediment and the fluid composition remain invariant, and
2) sufficient number of suspended particles are available for promoting
coagulation and aggregation.
The determination of c^ and corresponding to each sedimentfluid
mixture can be carried out through rheological diagrams of applied shear
stress against the shearing rate. Such plots were developed by Krone
(1963; 1978) with the help of a specially designed annular viscometer.
An example of such a diagram is presented in Fig. 2.12, with the shear
stress proportional to the dial reading on the viscometer and the
shearing rate proportional to the rotation rate of the outer cylinder of
the viscometer. Each order of aggregation corresponds to a given volume
fraction of the aggregates (volume occupied by the aggregates divided by
the total volume of the suspension) which in turn can be shown to be
related to the relative differential viscosity (the viscosity of the
suspension divided by the viscosity of the suspended medium). Given the
viscosity of the suspending medium, the relative differential viscosity
39
Q.
a '9Niav3H nvia
40
is determined from the slope of the rheological diagram, and hence the
volume fraction can be calculated. The density is then computed from
the volume fraction. The intercept on the applied shear stress axis of
the diagram corresponds to i;^; in Fig. 2.12 the ordinate intercept is
proportional to t^^. Table 2.1 gives the orders of aggregation, cation
exchange capacity (CEC), densities and shear strengths of sediment
samples from five different sources. As observed in this table, the
first four sediment samples are characterized by three orders of
aggregation while the sediment from San Francisco Bay is characterized
by six orders. The number of aggregation orders possible for a
suspension of a given sediment is equal to the number of linear segments
on the rheological diagram with different slopes. Thus, in Fig. 2.12,
the sediment sample has two possible orders of aggregation. Krone
(1963) postulated that each segment is related to a particular volume
fraction and therefore to a different manner in which the same sediment
can aggregate, i.e. different order of aggregation. Thus, for the
suspensions of the first four sediments listed in Table 2.1, three
different linear segments were obtained on the rheological diagrams,
while for Bay mud, six segments, and therefore six orders of aggregation
were found. This indicates that Bay mud can aggregate in three more
ways than the other four sediments, and further suggests that the Bay
sediment is more cohesive than the others. Also observed in this table
is the very rapid decrease in the shear strengths and somewhat less
rapid decrease in densities with increasing order of aggregation. These
trends indicate that as the order of aggregation increases, the inter
aggregate pore volume increases and the strength of these aggregates
decreases because of limited bonding area between the lower order
aggregates (Krone, 1978).
41
Table 2.1
Properties of Sediment Aggregates (after Krone, 1963)
Sediment Order of CEC Density Shear Strength
Sample Aggregation (meq/100 gm) "^"^^ ^s^'^ ^'^^
Brunswick
Harbor 0 38 1164 3.40
1 1090 0.41
2 1067 0.12
3 1056 0.062
Wilmington
District 0 32 1250 2.10
1 1132 0.94
2 1093 0.25
3 1074 0.12
Gulf port
Channel 0 49 1205 4.60
1 1106 0.69
2 1078 0.47
3 1065 0.18
VJhite River
(salt) 0 60 1212 4.90
1 1109 0.68
2 1079 0.47
3 1065 0.19
San Francisco
Bay 0 34 1269 2.20
1 1179 0.39
2 1137 0.14
3 1113 0.14
4 1098 0.082
5 1087 0.036
6 1079 0.020
42
The settling rate of coagulated sediment particles depends on, in
part, the size and density of the aggregates and as such is a function
of the processes of coagulation and aggregation (Owen, 1970). Therefore
the factors which govern these two processes also affect the settling
rate of the resulting aggregates. As noted in Chapter I, the settling
velocities of aggregates can be several orders of magnitude larger than
those of individual clay particles (Bellessort, 1973).
Deposition of aggregates occurs relatively quickly during slack
water. Deposition also occurs in slowly moving and/or decelerating
flows, as was observed (see Fig. 2.13), for example, in the Savannah
River Estuary during the second half of flood and ebb flows (Krone,
1972). Under such conditions only those aggregates with shear strengths
of sufficient magnitude to withstand the highly disruptive shear
stresses in the near bed region will actually deposit and adhere to the
bed. Thus, deposition is governed by the bed shear stresses, turbulence
structure above the bed, type of sediment, depth of flow, suspension
concentration and the ionic constitution of the suspending fluid (Mehta
and Partheniades, 1973). An important conclusion derived from extensive
laboratory erosion and deposition experiments using a wide range of
cohesive sediments under steady flow conditions was that under these
conditions the two processes do not occur simultaneously as they do in
cohesionless sediment transport (rlehta and Partheniades, 1975; 1979;
Parchure, 1983).
A flowdeposited bed of cohesive sediment aggregates possesses a
vertical bulk density and shear (i.e. yield) strength profile which
changes in time primarily due to consolidation. Secondary causes are
thixotropy and associated physicochemical changes affecting
43
interparticle forces. Consolidation, caused by the gravitational force
(overburden) of overlying deposited aggregates which crushes and thereby
decreases the order of aggregation of underlying sediment, has been
observed to occur in three phases (Migniot, 1968). During the first
phase the bed consolidates quickly as the water in the bed moves upward
through the interstices of the bed material. This phase has been found
to last up to approximately 10 hours for cohesive sediments (Owen,
1977). During the second phase, which can last up to about 500 hours,
water is expelled from the bed by percolation. The rate of
consolidation during the third phase is even slower and the length of
time it takes for a cohesive sediment bed to reach its final, fully
consolidated state depends upon the nature of the sedimentary material
comprising the bed and the chemical composition (i.e. ionic
concentrations) of the bed pore water (Owen, 1977). The average values
of the bed bulk density and shear strength increase and their vertical
profiles change during each of these three phases. Consideration of the
consolidation process is essential in modeling the erosive behavior of
such beds because: 1) the susceptibility to erosion of a consolidating
bed decreases with time due to the continual increase in shear strength,
and 2) the vertical profile of the shear strength determines the level
to which a bed will erode when subjected to excess shear, i.e. an
applied bed shear stress in excess of the shear strength of the bed
surface.
From an Eulerian point of view, the superposition of oscillating
tidal flows on the quasisteady state transport phenomenon depicted in
Fig. 1.1 results in corresponding oscillations of the suspended sediment
concentration with time as shown by the Savannah River data in Fig.
44
2.13. Such a variation of the suspended load uUimately results from a
combination of advective and dispersive transport, erosion and
deposition.
Because of the complexity of the phenomena, more than one
interpretation is possible as far as any schematic representation of
these phenomena is concerned. One such representation is shown in Fig.
2.14. According to this description, cohesive sediments can exist in
four different physical states in a tidal estuary or sea: as a mobile
suspension, a stationary suspension, a partially consolidated bed and as
a settled bed. The last two are formed as a result of consolidation of
a stationary suspension. Stationarity here implies little horizontal
movement, although consolidation does mean that there is vertical
(downward) movement. A stationary suspension, a partially consolidated
bed and a settled bed may erode if the shear stress exceeds a certain
critical value. Erosion of a stationary suspension is referred to as
redispersion while erosion of a partially consolidated as well as a
settled bed is termed resuspension.
45
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05
CHAPTER III
SEDIMENT TRANSPORT MECHANICS
3.1 Introductory Note
The purpose of this chapter is 1) to discuss the mechanics of
cohesive sediment transport and 2) to describe the algorithms developed
during this investigation. The processes for which algorithms have been
developed include erosion, dispersive transport, deposition, bed
formation and subsequent consolidation. The chapter begins with a
description of the equations which govern the depthaveraged, uncoupled
movement of water and suspended cohesive sediments.
3.2 Governing Equations
3.2.1 Coordinate System
A righthanded Cartesian coordinate system is used (Fig. 3.1). The
positive Xaxis is coincident with the longitudinal axis of the estuary
and points downstream. The coordinate system origin is located at some
datum below the bed level. The positive zaxis is the vertical
dimension and points upward. The yaxis defines lateral distances and
points from right to left.
3.2.2 Equations of Motion
The equations which govern the twodimensional, depthaveraged
unsteady turbulent movement of an incompressible viscous fluid are
statements which express two of the basic principles of Newtonian
47
48
49
physics, that of conservation of mass (continuity equation) and the
conservation of momentum (equations of motion). These equations are
solved numerically in order to describe the velocity field in the
estuary or other water body of interest. Alternately, the velocity
field may be measured in a physical scale model of the estuary.
However, in this study of the "uncoupled" movement of cohesive
sediments, these governing equations are not solved as it is assumed
that the velocity field is known beforehand. The continuity equation
and the two equations of motion are included and discussed here for the
sake of completeness.
3.2.2.1. Continuity
The conservation of mass, as expressed by the continuity equation,
states that the mass of an incompressible fluid entering a control
volume per unit time is equal to the sum of the fluid mass leaving the
control volume plus the change in volume of the control volume. The
depthaveraged continuity equation for an incompressible fluid is
ad a a
— +— (u.d)+ (v.d) = 0 (3.1)
at ax ay
where d = depth of flow and u, v = time and depthaveraged water
velocity components in the x and and y  directions, respectively.
3.2.2.2. Conservation of Momemtum
The conservation of momentum (which is Newton's second law of
motion) for an incompressible fluid states that the product of the fluid
mass and acceleration is equal to the sum of the body (gravitational)
50
forces and the normal (pressure) and tangent (friction) surface forces
which act on the boundaries of the water body. The twodimensional,
depthaveraged equations of motion for an incompressible viscous fluid,
which can be derived from the NavierStokes equations, are given by
Qu 5li Qu 1 1 5 a
— + u — + V — = + — [ — h — ^ 1 + 2wvsin<j) 
at dx ay dx 5x ay
(uSv^)/2+ _L_a cos(e)
C^d p d
w
av av av i ap i a 5
_+u— +v— = + — [— T + — T 1  2oJusin* +
at ax ay p^ dy p^ dx ay
(3.2)
2 2I/0 Pa'^a 2
— (u +v^)^2+^V%sin(e) (3.3)
8d p d
w
az
in which:
p = pressure force
P^^ = fluid density
Pg = air density
■^■jj = horizontal turbulent shear stresses
w = angular velocity of the earth
<t> = local latitude
Ufa*
51
g = acceleration due to gravity
f = DarcyVieisbach friction factor
Vg = wind speed at a reference elevation
above the water surface
0 = angle between the wind direction and the
positive Xaxis
= wind drag coefficient
Equation 3.4 is the hydrostatic equation which results when the vertical
component of the flow velocity and acceleration are small relative to
the horizontal flow velocity and acceleration. The third term on the
right hand side of Eqs. 3.2 and 3.3 is the Coriolis acceleration in the
Northern hemisphere in the x and y directions, respectively. The
fourth and fifth terms on the right hand side of Eqs. 3.2 and 3.3
represent the effects of bottom shear stresses and surface wind shear
stresses in the x and y directions, respectively. The three terms on
the left hand side of Eqs. 3.2 and 3.3 represent the substantive fluid
acceleration in the x and y directions, respectively.
3.2.3. Advection  Dispersion Equation
The principle of conservation of mass with appropriate source and
sink terms describes the advective and dispersive transport of suspended
sediment in a turbulent flow field. In this law, expressed by the
advectiondispersion equation, the timerate of change of mass of
sediment 1n a stationary control volume is equated to the spatial rate
of change of mass due to advection by an external flow field plus the
spatial rate of change of mass due to diffusion and dispersion
52
processes. Both the threedimensional form and the twodimensional,
depthaveraged forms of the advectiondispersion equation are derived in
Appendix A. The latter is given here:
C = mass of sediment per unit volume of water
and sediment mixture
D^j = effective sediment dispersion tensor
Sj = source/sink term.
Implicit in Eq. 3.5 is the assumption that the suspended material is
advected in the x and y directions at the respective water velocity
components. This assumption is reasonable for sediment that is not
transported as bed load since rolling and saltation of the sediment,
which occurs during bed load transport, can cause a significant
difference between the water and sediment velocities. Sayre (1968)
verified that this assumption is approximately true for sediment
particles less than about 100 m in diameter. The source/sink term in
this equation can be expressed as
9 5 a d ac ac
— (dC) + u — (dC) +v — (dC) = — {dD — + dD — } +
at ax ay ax ^^ax ^^ay
a ac ac
I — + dD — } +
y'^ax yyay ^
(3.5)
where:
dC dC
S = (— I + —I
' dt^ dt'^
(3.6)
L
53
dC,
where — L is the rate of sediment addition (i.e. source) due to erosion
dC
from the bed, and —l^j is the rate of sediment removal (i.e., sink) due
. . dC, dC
to deposition of sediment. Expressions for — L and — L are qiven
dt ^ dt^
respectively in Sections 3.4.3 and 3.6.3. S[_ accounts for the removal
(sink) of a certain mass of sediment, for example, by dredging in one
area (e.g. navagational channel) of a water body, and the dumping
(source) of the sediment as dredge spoil in another location in the same
body of water.
In the following section, the schematization for sediment beds is
described. This description is preceded by a general discusson on the
nature (i.e. structure) of these beds as revealed in several laboratory
investigations.
3.3 Sediment Bed
3.3.1 Bed Structure
Surficial layers of estuarial beds, typically composed of flow
deposited cohesive sediments, occur in three different states:
stationary suspensions, partially consolidated (or consolidating) beds
and settled (or fully consolidated) beds. Stationary suspensions are
defined by Parker and Lee (1979) as assemblages of high concentrations
of sediment particles that are supported jointly by the water and the
developing skeletal soil framework, and which have no horizontal
movement. These suspensions, which may be regarded as extremely under
consolidated soil, develop whenever the settling rate of concentrated
mobile suspensions exceeds the rate of selfweight consolidation (Parker
and Kirby, 1982). They tend to have a high water content (therefore low
bulk density) and a very low, but measurable, shear strength, v, ^^^^
54
must be at least as high as the bed shear, v^, which existed during the
deposition period (Mehta et al_., 1982a). Thus, they exhibit a definite
nonNewtonian rheology. Kirby and Parker (1977) found that stationary
suspensions have a surface bulk density of approximately 1050 kg/m^ and
a layered structure. Krone (1963) found that, in addition to the bed
shear, the structure (or framework) of these suspensions depends on the
aggregate order in the following manner: if the aggregates deposit
without being broken up by the bed shear, the surficial layers of these
suspensions will be composed of an aggregate network whose order is one
higher than that of the individual settling aggregates; therefore, these
layers will have lower bulk densities and shear strengths than those of
the aggregates which form them.
Whether or not entrainment of these suspensions, also referred to
as redispersion (Parker and Kirby, 1977) and mass erosion (Paaswell,
1973), occurs during periods of erosion depends upon the mechanical
shear strength (i.e. stability) of this aggregate network. That portion
which remains on the bed undergoes: 1) selfweight consolidation, due
to overburden pressure resulting from the weight of the overlying
sediment which crushes the aggregate network below, and 2) thixotropic
effects, defined as the slow rearrangement of deposited aggregates
attributed to internal energy and unbalanced internal stresses
(r^itchell, 1961), both of which reduce the order of aggregation of the
subsurface bed layers. This implies that the bed becomes stratified
with respect to bulk density and shear strength, with both properties
typically increasing montonically with depth, at least under laboratory
conditions (Mehta et ail_., 1982a). Stationary suspensions generally have
a lifespan that varies from a few hours to a few days. Differential
55
settling caused by sorting processes is another cause of stratified bed
formation.
Continued consolidation eventually results in the formation of
settled mud, defined by Parker and Lee (1979) as "assemblages of
particles predominantly supported by the effective contact stresses
between particles as well as any excess pore water pressure." This
portion of the bed has a lower water content, a lower order of
aggregation, and a higher shear strength and therefore is better able to
resist high bed shear stresses. The settled mud in the Severn Estuary
and Inner Bristol Channel, United Kingdom, has a bulk density range from
1,300 to 1,700 kg/m^ (Kirby and Parker, 1983).
In this study the primary characteristic used to distinguish
between a stationary suspension and a partially consolidated or settled
bed is the mode of failure that occurs when the surface of the
suspension or bed is subjected to an excess shear stress (i.e. i^i^ >
t;^.). Erosion by particle by particle or aggregate by aggregate removal
is not a correct representation in areas where stationary suspensions
exist (Kirby and Parker, 1983). As stated previously, stationary
suspensions undergo redispersion while partially consolidated and/or
settled beds undergo resuspension (Parker and Kirby, 1977) or surface
erosion (Paaswell, 1973). Both erosion processes are discussed in
Section 3.4.1.
The nature of the density and shear strength profiles typically
found in flowdeposited cohesive sediment beds has been revealed in
laboratory tests by, among others, Richards _et_^. (1974), Owen (1975),
Thorn and Parsons (1980), Parchure (1980), Bain (1981) and Dixit
(1982). A review of this subject is given here.
56
Figure 3.2 shows the dimensionless density profile measured by Owen
(1970) after 4.2 and 8.3 hours of consolidation for mud obtained from
the Thames near Dagenham, England. The indeterminate effect of salinity
on the density profile for two different beds after 67 hours of
consolidation for mud obtained at the entrance to the Royal Edwards
Docks, Avonmouth, Bristol, England is seen in Fig. 3.3. In fact,
despite the difference in mean density the same dimensionless profile is
drawn through the data points in both Figures 3.3(a) and 3.3(b). The
only distinguishable difference between these two sets of density
profiles is the slightly lower relative surface densities in the bed
formed by settling of the lower initial suspension concentration of 7.72
g/1. The Avonmouth mud is composed predominantly of illite, which is a
relatively inert clay mineral, and was found to have a CEC value of 17
meq/100 gm. Therefore, it is not surprising that salinity had very
little effect on the bed density.
Figure 3.4 shows the measured dimensionless density profiles for
four different bed thicknesses after 67 hours of consolidation for the
Avonmouth mud (Owen, 1970). The same average profile drawn in Figure
3.3 was drawn on this figure as well. A remarkable fit obtained between
the data and this common dimensionless profile is evident.
Figure 3.5 shows the dimensionless density profiles obtained by
Thorn and Parsons (1980) after two days of consolidation for muds from
the Forth Estuary at Grangemouth, Scotland, the Brisbane River at the
Port of Brisbane, Australia, and the dredged channel to the Port of
Belawan, Sumatra, Indonesia. The percentage of clay minerals,
percentage of nonclay minerals and the cation exchange capacity for the
Grangemough mud, Brisbane mud and Belawan mud were 51%, 50% and 7580%
57
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58
59
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60
(clay minerals), 39%, 50% and 20% (nonclays) and 20 meg/100 gm, 35
meg/100 gm and 25 meg/100 gm (CEC), respectively. Thus, the Grangemouth
mud is the least cohesive and the Brisbane mud is the most cohesive.
This is not unexpected as the Brisbane mud has the highest percentage of
montmorillonite (approximately 60% of the clay mineral fraction), which
is a very active (cohesive) clay mineral.
Figure 3.6 shows the dimensionless density profiles (normalized
with respect to the initial suspension concentration) measured by
Parchure (1980) for commercial grade kaolinite after 24, 40 and 135
hours of consolidation. This clay had a CEC of approximately 9
meq/100 gm.
Figure 3.7 shows the dimensionless density profile for mud from the
Mersey Estuary, England after 48 hours of consolidation (Bain, 1981).
This mud was composed of 76% clay minerals and 24% silica.
Figure 3.8 shows dimensionless density profiles of two different
beds of the same natural mud after 48 hours of consolidation (Thorn,
1981). The solid line profile was measured using a nuclear
transmissometer, while the discrete point profile was determined using a
layerbylayer sampling technique.
Figure 3.9 shows the dimensionless density profiles found by Dixit
(1982) for flowdeposited beds of commercial grade kaolinite after 2, 5,
11 and 24 hours of consolidation while Figure 3.10 shows these profiles
after 48, 72, 96, 144 and 240 hours of consolidation.
It is evident from all these density profiles that a static or
dynamic deposited cohesive sediment bed has 1) a characteristic
elongated^ slightly reversed S shape density profile that generally
increases monotonically with depth and that is independent of the bed
61
Fig. 3.6. Variation of Bed Density v/ith Depth for Three Different
Conditions of Flow Deposited Beds (after Parchure, 1980).
62
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63
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54
thickness and 2) a very Tow, generally indeterminate surface density.
The variation of the density profile with consolidation time is examined
in Chapter III, Section 3.7.
Parchure (1980) made the following observation with regard to the
shear strength profiles, in flowdeposited cohesive sediment
beds: '^^(z) increases rapidly with distance below the watersediment
interface for z < i^^, at which i^ = t^j^, where t^i^ (defined in Section
3.4.1) is a characteristic value of r^. For z > z^^, ^^^(z) continues to
increase but at a greatly decreased rate (Figure 3.11). The influence
of such a '^(iz) profile on the erosion rate is discussed in the
following section. Figure 3.12 shows a '^(^iz) profile found by Dixit
(1982). From the tests conducted by Dixit, the following two
observations may be made: 1) such a "^(.iz) profile was not found in five
out of nine experiments, and 2) the sediment beds used by Dixit were up
to six times thicker than those used by Parchure. Therefore, the t^(.(z)
profiles measured by Dixit are naturally more representative of
estuarine beds, and as such it is believed that until further studies
are conducted, no definitive statement regarding the precise nature of
"^(.{z) profiles in cohesive sediment beds can be made. The possibility
of a correlation between the bed density and shear strength of cohesive
sediment beds is examined in Section 3.7.
3.3.2. Effect of Salinity on Bed Structure
For most cohesive soils the interparticle and interfloc contact is
considered to be the only significant region between particles where
normal stresses and shear stresses can be transmitted (Mitchell et al..
65
E
£
u
o
<
Li.
tr
UJ
H
Expt. 17
=0.05 H/m
T^,= 24 hrs
Tch = 0.2l N/m^
hJ
<
I
I
u
o
UJ
cn
$
O
J
LU
OQ
X
o.
Q
Expt. 18
Tb =0.015 N/m^
V 40 hrs
T. =0.28 N/m'
cn
Expt. 19
Tt, = 0 N/m'
T^= I35tirs
T^h=0.34N/m
Fig. 3.11.
BED SHEAR STRENGTH (N/m^)
Bed Shear Strength Profiles for Kaolinite Beds
(after Parchure, 1980).
, — — —
Fig, 3.12. Bed Shear Strength Profile for a Kaolinite Bed (after
Dixit, 1982).
67
1969). In particular, it seems very likely that the primary role of the
doublelayer interaction and other physicochemi cal forces is to control
the structure of the soil and to alter the transmitted stresses from
what they would be due to the applied flowinduced shear and overburden
normal stresses alone. Two factors that effect the structure of a
cohesive soil, swelling and permeability, and the effect salinity has on
these factors are discussed next.
The degree of swelling which occurs when a soil is immersed in a
fluid is Influenced by factors such as the amount of clay, shape and
size of the particles, the salinity and the sodium adsorption ratio
(SAR) of the eroding and pore fluid, and the presence of an imposed load
on the swelling areas (Grimshaw, 1971). Sargunam et aK (1973) state
that decreases in the salinity of the eroding fluid or increases in the
SAR cause the surface clay particles to swell more. This swelling
causes a weakening of the interparticle attractive forces and thus
increases the susceptibility of the soil to erosion. Increasing the
salinity of the eroding fluid causes a greater compression of the
diffuse layer, thereby reducing the repulsive forces of soil
particles,. This reduction serves to limit the amount of swelling.
Sargunam et_^. (1973) found that when the salinity of the eroding
fluid is greater than that of the pore fluid, the yield strength of the
soil is greater and therefore the erosion potential is decreased. In
this case the osmotic pressure gradient across the fluidbed interface
may result in deswelling, or consolidation of the bed sediment
particles, which would cause an increase in the interparticle bonding
forces and therefore lessen the susceptibility to erosion.
68
It 13 believed that while this phenomenon of swelling influences to
some degree the structure and hence the erosion potential of a cohesive
bed, it is not nearly as significant as the upward flux of pore water
due to gravitational forces in a consolidating mud.
Quirk and Schofield (1955) found that the degree of permeability of
clay soils depends upon the nature and the concentration of the cations
present in both the eroding and pore fluids. In particular they found
that permeability increased with an increase in the salinity of the
eroding fluid. Swelling, stabilization (i.e. decoagulation) and
consolidation are generally considered to be the main reasons for
changes in the permeability. The former can cause either partial or
total blockage of soil pores which would result in a decrease in
permeability. Stability essentially occurs during swelling when the
clay particles have separated to the extent that the interparticle
repulsive forces are dominant over the attractive forces. Since an
increase 1n the salinity of the eroding fluid serves to limit the amount
of swelling which occurs and thus restricts the amount of stabilization
as well, an increase in salinity would result in increased
permeability. The converse was found to occur as well since, as stated
previously, a decrease in the salinity of the eroding fluid causes an
increase in the degree of swelling and stabilization.
As mentioned previously, the effect the salinity of the pore fluid
has on the bed density could be expected to be a direct function of the
cation exchange capacity of the sediment. Salinity was seen (in Figure
3.3) to have very little effect on the density profile for the
relatively inert Avonmouth mud. Figure 3.13 shows the indeterminate
effect of salinity on the bed density profile for mud from Lake Francis,
69
70
Nebraska, of which 50% was finer than 2 m (claysized particles), with
montmorinonite, illite, kaolinite and quartz being the predominant
minerals, and with a CEC of 100 meq/ion gm. This high CEC value
indicates a higher percentage of montmorillonite than the other two clay
minerals. Evidently, salinity and the CEC value had very little effect
on the bed density profile.
Using the method described, by Mehta et_al_. (1982a), the bed shear
strength (or the critical shear stress for erosion), t^, of Lake Francis
mud as a function of depth below the initial bed surface, z^, was
determined as a function of salinity for salinities from 0 to 10 ppt
(Figure 3.14). Two trends are observed in this graph. First,
increases with depth in the upper part of the bed for all salinities (no
definite data could be obtained for the lower part of the bed, i.e. for
00 0.2 0.4 0.6 0.8
BED SHEAR STRENGTH, (N/m^)
Fig, 3.14. Bed Shear Strength Profiles as Functions of Salinity.
71
> 0.5 cm, inasmuch as this portion of the bed did not erode during
these experiments). Second, increases with increasing salinity from
0 to 2 ppt; thereafter, for salinities up to 10 ppt, no measurable
increase in '^^ occurred.
3.3.3. Bed Schematization
To facilitate the modeling of changes in the bed surface elevation
due to erosion, deposition and consolidation processes, the bed is
treated in the following manner: 1) it is discretized into a number of
layers and 2) the bed properties, e.g. thickness, are assumed to be
spatially (in the xy plane) invariant within each element, but not so
from element to element, in order to account for interelement spatial
variances in shoaling and/or scouring patterns. These two factors are
expounded upon below.
The bed in each element is considered to be composed of two
sections: 1) the original, settled (consolidated) bed that is present
at the start of modeling and 2) new deposits located on top of the
original bed, that result from deposition during the modeling. Each of
these two sections is divided into a number of layers in order to
specify the actual shear strength and bulk density profiles in the
model. The new deposit bed section is subdivided into two subsections,
the top referred to as unconsolidated new deposit (UNO) layers and the
bottom as partially consolidated new deposit (CND) layers (Figure
3.15). The former subsection, i.e. the one corresponding to a
stationary suspension, is considered to undergo redispersion while the
latter, i.e. the partially consolidated bed, undergoes resuspension when
subjected to an excess shear stress. The settled bed as well undergoes
72
©
Unconsolidated
New Deposit
(UND)
Partially (2)
Consolidated ^
New Deposit
(PCND) ^
®
®
®
Tu
Bed Surface
TuNDJ^TLAYMd))
^ '
T
UND, 2
NLAYTM = 3
T,
UND, 3
TcND.i (TLAY(I))
New Deposits
CND,2
CND, 3
NLAYT=4
T,
CND, 4
To, I (THICKO(I))
T
0,2
Settled Bed
T
0,3
NLAY0=4
To,4
Fig. 3.15. Bed Schematization used in Bed Fonnation Algorithm.
Fig. 3
BED SHEAR STRENGTH , ( N/m^)
0.2
16.
Hypothetical Shear Strength Profile Illustrating
Determination of Bed Layers Thicknesses.
73
resuspension. The number of layers indicated in Fig. 3.15 for each of
the three bed sections are not fixed, as each section can be assigned
any given number of layers.
Stationary suspensions are represented in the depthaveraged model
as being the top section of the layered bed model, even though they are
not a true bed or soil, in order to account for the subsequent
redispersion and/or consolidation of these suspensions. However, the
time varying thickness of the bed in each element is equal to the sum of
only the NLAYT CNO layers and the NLAYO settled bed layers. The
following bedrelated parameters are required for bed schematization in
the model :
1) The bed shear strength profile in the UND. This can be ascertained
from laboratory erosion tests using samples of the sediment from
the water body being modeled (see Appendix D, Section D.2).
2) The number of UND layers (NLAYTM) and the thickness of each layer
(TLAYM(I), 1=1, MLAYTM). These parameters must be determined using
the shear strength profile. For example. Figure 3.16 shows a
hypothetical t^(z) profile and illustrates that this bed section
must be divided such that the variation of within each layer is
approximately linear. The values at the NLAYTM+1 nodes need to
be read into the model.
3) The dry sediment density values at the NLAYTM+1 nodes as well need
to be determined. The p(z) profile may be determined using a
laboratory freezedryi ng method (appropriately modified for field
samples, where necessary) described by Parchure (1980), the pumping
method described by Thorn and Parsons (1977), a gammaray nuclear
transmission densitometer (Whitmarsh, 1971) or a nondestructive X
ray technique (Been and Sills, 1981).
74
4) The same parameters for the CND layers and the settled bed layers
must be determined, with the settled bed parameters determined for
each element (where an original bed exists). The parameters read
in for the UND and CND layers are assumed to be constants for all
elements.
5) A stationary suspension and/or partially consolidated bed present
on top of the settled bed at the start of the modeling is simulated
by reading in the dry sediment mass per unit bed area of such new
deposits obtained from every element where such exist.
The bed level at which the dry sediment density is approximately
480 kg/m^ is usually taken to be the top of the settled bed. Thus, the
sediment located above this level is considered to be new deposits.
Another method which may be used to differentiate between new deposits
and the settled bed is described in Appendix D, Section D.2.
Included in Appendix D is a brief description of how these various
bed properties can be determined through the use of a field data
collection program and a laboratory testing program. Other parameters
characterizing the rate of resuspension that each layer undergoes when
subjected to an excess shear must be evaluated as well; these are
discussed in Section 3.4.3.
The following procedure was developed for forming the new deposit
bed layer(s) which result from new deposits initially present on top of
the settled bed and/or deposited during the modeling.
The dry sediment mass per unit bed area per element, Mg, read in
initially if a new deposit exists in any given element, or deposited
during modeling (as determined by the deposition algorithm) is used in
conjunction with the UND and CND properties to solve iteratively for the
75
thickness of bed formed by Mq for each element where Mq > 0. This
thickness, at, depends on the dry sediment density profile, p(z), for
the UND and CND layers. The thickness AT is determined using the
following relationship:
J* P(2)zdz
AT = ~ (3.7)
where * = AT ± 0.02AT. If AT is greater than TLAYM(l) (see Figure 3.15)
then more than one layer of UND is added. The assumed linear variation
of p within each layer is used in the above equation. When or if the
UND layers are filled, the same procedure is used to fill up the CND
layers below the UND layers. The bottom CND layer can never fill up;
therefore, continuing deposition is accounted for by increasing the
thickness of this layer, while the thicknesses of the overlying UND and
CND layers remain the same. This particular filling sequence was used
in order to account for the consolidation of the sediment bed due to
overburden pressure during the bed formation phase by virtue of the
increasing and p values with bed depth.
In the next section, a discussion on the erosive behavior of fine
sediment beds is given, followed by a description of the erosion
algorithm.
3.4 Erosion
3.4.1. Previous Investigations
Of interest in this study are the erosion (resuspension)
characteristics of saturated, flowdeposited cohesive sediment beds. A
number of laboratory investigations were carried out in the sixties and
76
early seventies in order to determine the rate of resuspension, e,
defined as the mass of sediment eroded per unit bed surface area per
unit time, as a function of a bed shear stress in steady, turbulent
flows. An important conclusion from these tests was that the usual soil
indices such as liquid or plastic limit do not adequately describe the
erosive behavior of these soils (Mehta, 1981). For example,
Partheniades (1962) concluded that the bed shear strength as measured by
standard tests, e.g. the directshear test (Terzaghi and Peck, 1960),
has no direct relationship with the soil's resistance to erosion, which
is essentially governed by the strength of the interparticle and inter
aggregate bonds between the deposited sediment material. Shown in Table
3.1 are various physicochemical factors known to govern the erosive
properties of these beds. These factors must be specified to properly
characterize the erosive behavior. The hydrodynamic factors define the
erosive forces while the bed and fluid physicochemical properties
determine the resistivity of the bed to erosion.
The erosive forces, characterized by the flowinduced instantaneous
bed shear stress, are determined by the flow characteristics and the
surface roughness of the fluidbed interface. The sediment composition,
pore and eroding fluid compositions and the structure of the flow
deposited bed at the onset of erosion must be determined in order to
properly define the erosion resistance of the bed. Sediment composition
is specified by the grain size distribution of the bed material (i.e.
weight fraction of clays, silts), the type of clay minerals present, and
the amount and type of organic matter. The CEC can be used to
characterize clay composition using the apparent dielectric constant
measured at selected frequencies. Each clay tested appeared to have a
77
Table 3.1
Principle Factors Controlling Erosion of
Saturated Cohesive Sediment Beds
HYDRODYNAMIC FACTORS (Erosive Force)
BED SHEAR STRESS
'Flow Characteristics
•Bed  Fluid Interface
BED AND FLUID PROPERTIES (Resistive Force)
SEDIMENT COMPOSITION
PORE FLUID
COMPOSITION
ERODING FLUID
COMPOSITION
■Clay Mineral Type] Ion Exchange Capacity
•Clay Percentage by Weight
■Organic Matter
■ Monoand Divalent Cations Concentrations ] Conductivity
Relative Abundance of \ car l(Na"*",Ca"^.Mg^)
Monoand Divalent Cations/ ^'^'^ v. , . y
Temperature
pH
Salinity (NaCI,CaCl2,MgCl2)
■Temperature
•pH
Cementing Agents (Iron Oxide, etc)
BED STRUCTURE
/Placed Bed
■Stress History I D3pQ3i^3d
78
characteristic value of a "dielectric dispersion" parameter determined
from these measurements. The dielectric dispersion has been defined as
the amount of decrease in the apparent dielectric constant with
frequency (Alizadeh, 1974).
The composition of the pore and eroding fluid are specified by the
temperature, pH, total amount of salts and the type and abundance of
ions present, principally CI", Na+, Ca^^, and Mg2+. Cementing agents
such as iron oxide can significantly increase the resistance of a
sediment bed to erosion. Measurement of the electrical conductivity is
used to determine the total salt concentration. The physicochemical
aspects pertaining to the aforementioned factors have been summarized by
Sargunam et _al_. (1973), Kandiah (1974), Arulanandan et_ al_. (1975) and
Ariathurai and Arulanandan (1978). The effect of the bed structure,
specifically the vertical dry sediment density and shear strength
profiles, on the rate of erosion is discussed by Lambermont and Lebon
(1978) and f^ehta et al_. (1982a).
Several different types of relationships between the rate of
erosion, £, and the timemean value of the flowinduced bed shear
stress, Xj^, have been reported for nonstratified beds. These include
statisticalmechanical models (Partheniades, 1965; Christensen, 1965), a
rate process model (Paaswell, 1973; Kelly and Gularte, 1981) and
empirical relationships (Ariathurai and Arulanandan, 1978). These
relationships typically have the following general form (Mehta, 1981):
(3.8)
where ili,^!^, . . . ,ti^. are parameters that specify the bed resistivity.
79
The resuspension rate, £, is related to the timerate of change of
the suspension concentration, dC/dt, and to the timerate of change of
the depth of erosion, Zj^, with respect to the original bed surface
elevation, by the following expressions:
dC
e = d— (3.9)
dt
dC 1 dz.
— = p(z.)— (3.10)
dt d ° dt
Figure 3.17 shows the general nature of laboratory determined
relationships found by, among others, Partheniades (1962) and
Christensen and Das (1973) for placed beds. Placed beds are sediment
beds that are formed artificially. Such beds include those that are
remolded and/or compacted after placement in a test apparatus (Mehta and
Partheniades, 1979). In these beds, the shear strength and density
profiles show much less significant stratification over the depth of the
bed than in flowdeposited beds (Mehta et_ al_. , 1982a). Shown in Figs.
3.18 and 3.19 are examples of this relationship, which may be expressed as
^ = ^^^V^ch^ ^ ^ch ^'''^
80
Bed shear stress
Fig. 3.17. Laboratory Determined Relationship Between Erosion Rate, e,
and Bed Shear Stress, (after Mehta, 1981).
^ I 1 1 I 1 i 1 1 1 1 1
Fig. 3.18. Example of Relationship Betv/een e and x, (after Mehta, 1981).
81
Fig, 3.19. e  T^t) ^^^^ °^ Partheniades (1962), Series I and II
(after Mehta, 1981) .
82
where 'A = slope, and the subscript ch refers to a characteristic value
(flehta, 1981; Hunt, 1981). For x^ < x^^, M = and for t > t^^, M =
Mg. Thus, Eq. 3.11 has the general form z = e {x^, x^^, z^^, M^,
in Eq. 3.8. The parameter x^^^ determined by extrapolation of the M2
line to the £ = 0 axis, has been interpreted to be the critical shear
stress for erosion (Partheniades, 1962; Gularte, 1978). The
characteristic shear strength, t^^^ is defined to be the value of the
bed shear stress at which the M^^ and M2 lines intersect. Values of s^.^,
'''ch' '^1 ^^2 largely determined by the physicochemical factors
given in Table 3.1.
Ariathurai and Arulanandan (1978) found the same general
relationship for remolded beds as given in Eq. 3.11, but with = M^.
Thus, Eq. 3.11 becomes
b cr
£ = M (
cr
(3.12)
where M = M't^^^. Figure 3.20 gives an example of this relationship,
with ot^ = 1/m'.
Figure 3.21 shows the measured variation of C with time typically
found by several investigators (Partheniades, 1962; Mehta and
Partheniades, 1979; Mehta et_ al_. , 1982a) in laboratory resuspension
tests with flowdeposited (stratified) beds under a constant applied
■^5. As observed, dC/dt is high initially, decreases monotonically with
time and appears to approach zero. The value of x^ at the depth of
erosion at which dC/dt, and therefore e becomes essentially zero has
83
2.0
't5
Ariathurai and Arulanandan ( 1978)
30% illite
20
3.20. Dimensionless e  t Relationship Based on Results of
Ariathurai and Arulanandan (1978) (after Mehta, 1981)
1x1
2 4 6 8 10 12 14 16 18 20 22 24
TIME (Hours)
3.21. Relative Suspended Sediment Concentration Versus Time for
a Stratified Bed (after Mehta and Partheniades , 1979).
84
been interpreted to be equal to (Mehta et aT_. , 1982a). This
interpretation is based on the hypothesis that erosion continues as long
as \ > T^, i.e. the excess shear stress x^t^ > o. Erosion is arrested
at the bed level at which x^x^ = o. This interpretation, coupled with
measurement of p(z^) and the variation of C with t can result in an
empirical relationship for the rate of erosion of stratified beds.
Resuspension experiments with deposited (stratified) beds were
performed by Parchure (1980) in a rotating annular flume and by Dixit
(1982) in a recirculating straight flume. Both flumes are described in
Chapter V, Section 5.2. The objective of these experiments was to
determine the effect of varying bed shear strength with depth below the
initial bed surface on the rate of surface erosion under a flowinduced
shear stress. A description of the experimental procedures and results
from these experiments have been given by Parchure (1980), Dixit (1982)
and Mehta etal_. (1982a). A synopsis is given here.
A commercial grade kaolinite with a CEC of approximately 9 meq/100
gm was used in these experiments. Tap water, with a total salt
concentration of 0.28 ppt, pH = 8.5 and sodium adsorption ratio SAR =
0.012, was used in the recirculating flume, while tap water plus
commercially available sodium chloride at 35 ppt concentration, pH = 8.1
and SAR = 12.0 was used in the annular flume. The kaolinite was
equilibrated with the fluid for at least two weeks prior to the tests.
The equilibration time is an important factor that can affect the rate
of erosion due to the possibility of concentration gradients of ionic
constituents between the solid and the liquid phases, or between the
pore fluid and the eroding fluid, in the sedimentwater system if the
time allowed for equilibration is insufficient (Mehta, 1981).
85
The resuspension test methodology is depicted in Fig. 3.22.
Specifically, this figure shows how the bed shear was varied over the
course of each experiment. In Phase I the sedimentwater mixture with
sediment concentration was mixed at a high shear stress, i;^, for a
period T^. The shear stress t was greater than v , the maximum bed
max
shear stress at which deposition of suspended sediment occurs. In Phase
II the bed shear was lowered in steps, to for T^ , then to for
Tj^ and finally to zero shear stress for a period of consolidation,
J^^. During this phase the sediment settled out of suspension, formed a
bed and began to consolidate. As indicated in Fig. 3.22, the first two
phases define the preerosion stress history of the bed. In Phase III,
the shear stress was increased as shown in discretized (onehour) steps,
i.e. = T2 = ...T^. (= 1 hour), and resuspension of the deposited
material occurred.
The following parameters were held constant in most of the
experiments in each flume: C^, x^. T^, t^^, t^^, t^^' ^nd T^ .
Shown in Fig. 3.23 are typical values for these parameters and the
measured variation of suspended sediment concentration with time for a
test in tap water. The parameter At^. , which may be referred to as the
normalized incremental bed shear stress, is defined as (x. ^.xAfz.
1+1 1 1
where t. is the bed shear stress, z^, during the ith time step.
In steps i =1,...,5 in this figure it is apparent that the
suspension concentration approaches a constant value during the latter
stages of each timestep, i.e. dC/dt^, while for steps i = 5 and
especially 7 the values of dC/dt do not approach zero over the one hour
periods. This difference in the concentrationtime profiles is
represented in a different manner in Fig. 3.24, which shows the
86
0)
A
8?
T
i
rO
if
hr
_L
CVJ
CVJ
a
Jl
1
c
g
c
CD
Q.
CO
Z3
CO
<D
IT
V
O
CO
CD
+ 1
g
(/)
o
LU
I
CD
Q_
13
CO •
"O fO
O) CM
•r CO
I— CTl
Qrl
+J
O
fd
o ^
•r OI
+> s:
I i.
(O +>
fO
QJ
C
M
t. (/)
O O)
4 1—
>> C
O
O r
I (/I
o c
a oj
o Q.
to
+j zs
O) oo
2: cu
o;
a
OJ a
4) c
o ra
CD
r c
o
00
45
O) ra
J= S
+J ra
Q.
q QJ
o s
c
O T3
•. O)
+J CQ
n3
4 > CD
C C
<D •!
to S
CU 3
5 Q
Q.
O; to
a: to
O)
CJ s
■1 +>
e s
CU ro
^ OJ
O J=
00 00
CM
CVJ
ro
•I—
a.
87
\
E
CP
o
o
<
cr
H
u
o
o
00
z:
a.
CO
Z)
CO
3.50
11.00
10.00 Tr
ELAPSED TIME T (hrs)
4.50 5.50 6.50
7.50
6.00
0.00
'•00 2.00 3.00
ELAPSED TIME T (hrs)
4.00
Fig. 3.23. Variation of Suspension Concentration with Time for T =48
Hours (after Dixit, 1982). dc
88
 CQ = 44.i gm/Iiter
h =30.5 cm
I
Fig. 3.24. C(T^) Versus t^. for Three Values of T^^, Using Kaolinite
in Salt Water (after Mehta et al. , 1982a).
89
suspension concentration at the end of step i, C(T^. ), plotted against t.
for three different tests in the recirculating flume. The value of v^^,
a characteristic shear stress (similar to one defined previously for
tests with placed beds) is determined as shown for each test. It is
apparent that dC(T^)/dt is higher for t^ > t^^^ than for i^ < x^^. The
significance of this observation is better appreciated when it is
realized that £ is proportional to the excess shear stress, and
that increases more rapidly with depth, z^, for z^^ < z^,^, where z^^
is the depth below the initial bed surface at which = (Mehta et
al_., 1982a).
The following empirical relationship between and ■t:. = v^{z^) was
derived from these experiments:
e. = ^,_^exp[a..
1 c b
c b
(3.13)
where and <x. are empirical coefficients. Figures 3.25 and 3.26 show
this relationship for tests in tap water and salt water, respectively.
This relationship is analogous to the rate expression which results from
a heuristic interpretation of the rate process theory for chemical
reactions (Mehta et_ al_., 1982a). Christensen and Das (1973), Paaswell
(1973) and Kelly and Gularte (1981) have used the rate process theory in
explaining the erosional behavior of cohesive sediment beds. By
analogy, e^ is a quantitative measure of the work done by t^ on the
system, i.e. the bed, and e„ and a./i t, \ ... ^ ,
' 0^ '^■\/ '(.[z^] are measures of the system s
internal energy, i.e. bed resistance to an applied external force.
90
Fig. 3.25. Normalized Rate of Erosion, e./e.^. Versus Normalized Excess
Shear Stress, , Using Kaolinite in Tap
Water (after Mehta et a^. , 1982a).
91
Series 3
T(j(.= 40 hr
i
(g cmViin"')
• 1
0.100
5.9
0.04
o 2
0.120
5.5
0.25
^ 3
0.145
5.5
0.30
a 4
0.175
5.5
0.27
* 5
0.210
84
0.22
1 1 1 1
1.0
i.5
2.0
2.5
3.26. Normalized Rate of Erosion, e^/e^^. Versus Normalized
Excess Shear Stress, (t^/t^U^)) /t^U^) , Using
Kaolinite in Salt Water (after Mehta et al . , 1982a).
92
An important conclusion reached from the above experiments was that
new deposits should be treated separately from settled, consolidated
beds (Mehta et_ a1_. , 1982a). The rate of surface erosion of new deposits
may be evaluated using Eq. 3.13, while the erosion rate for settled beds
may be suitably determined using Eq. 3.12, in which £ varies linearly
with the normalized excess bed shear stress. The reasons for this
differentiation in determining e are twofold: 1) typical ^^ and p
profiles in settled beds vary less significantly with depth than in new
deposits, and may even be nearly invariant. Therefore, the value of
^\/^c^ " ^ = ^"^b relatively' small . For small values of At,^,
the exponential function in Eq. 3.13 can be approximated by a* (1 + Atj^)
which represents the first two terms in the Taylor series expansion of
exp{a(ATjj)). For small values of At^, i.e. Lz^ « i, both expressions
for £ vary linearly with Ac^^. Thus, the variation of £ with depth in
settled beds can be just as accurately and more simply determined using
Eq. 3.12. 2) The laboratory resuspension tests (briefly described in
Appendix D) required to evaluate the coefficients and a for each CND
layer can not be practically or easily performed using vertical sections
of the original settled bed (obtained from cores). A simpler laboratory
test has been described by Ariathurai and Arulanandan (1978) to evaluate
the variability of M with depth. This procedure is briefly noted in
Appendix D.
3.4.2. Effect of Salinity on Resuspension
Sherard_et^. (1972) have shown that the susceptibility of a
cohesive sediment bed to erosion depends on two factors: 1) the pore
fluid composition, as characterized by the SAR, and 2) the salinity of
93
the eroding fluid. It was found that as the eroding fluid salinity
decreases, soil resistivity to resuspension decreases as well. These
results were verified by Arulanandan et al_. (1975). In addition,
Sherard et a][. (1972) found that the erosion resistance decreased by
either the exchange of cations or a reduction of the valence of the
cations in the pore fluid. Kandiah (1974) and Arulanandan et al_. (1975)
confirmed these findings by showing that the erosion resistance
decreased and the rate of resuspension increased with increasing SAR
(and therefore decreasing valency of the cations) of the pore fluid.
Figure 3.27 shows such a relationship between the SAR and the critical
shear stress for erosion, which is a measure of soil resistance to
erosion (Alizadeh, 1974).
Experiments were conducted during this study to determine the
effect of the eroding fluid salinity on the rate of resuspension. The
experiments were performed in the rotating annular flume using the
bottom sediment from Lake Francis, Nebraska. Analysis of water from the
lake indicated the presence of Na"^, K"^, Ca^"^, Mg^^, Al^"^, Fe^"*", CI"
SO4 . These cations and anions would be expected to be present in the
sediment as well. The average pH of the lake water was 8.6. The
sediment was repeatedly washed in an attempt to remove these free salts
so that their effect on the sediment properties was minimized. The
washing was performed by immersing the sediment in deionized water,
vigorously stirring the sediment and water, allowing time for the
sediment to settle out of suspension by gravity, and then siphoning off
the clear supernatant water. This procedure was repeated at least three
times.
94
SODIUM ADSORPTION RATIO, SAR
Fig. 3.27. Critical Shear Stress Versus SAR for Montmorillonitic
Soil (after Alizadeh, 1974).
95
Commercial grade sodium chloride dissolved in different proportions
in tap water constituted the eroding fluid in these experiments. The
manufacturers of the sodium chloride supplied the data given in Table
3.2 regarding the contents of the processed sodium chloride. The cation
concentrations in sea salt, also included in this table, were obtained
from Bolz and Tuve (1976).
Table 3.2
Cation Concentrations in Processed
Sodium Chloride and Standard Sea Salt
Cation
NaCl
Sea Salt
Sodium
357460. ppm
301720. ppm
Calcium, Magnesium
50.
47770.
Potassium
10.
10860.
Phosphate
1.0
Iron
0.5
Tests were conducted for the following five salt concentrations:
0, 1, 2, 5 and 10 ppt by weight. However, as the concentrations of the
three most abundant cations, Na"^, Ca^"^ and Mg^'^, in the manufactured
salt were different from those in the standard sea salt (see Table 3.2),
the five different eroding fluids used in these experiments were not
exactly equivalent to sea water at the various salinities. In spite of
this, useful qualitative and quantitative information was obtained
regarding the effect of varying dissolved salt (i.e. electrolyte)
concentrations on the erosive characteristic of the mud.
I
96
The experimental procedures used in these tests has been described
in the previous section. Suspended sediment concentration as a function
of time as well as the bed density profile were measured. The values of
and Tj were 0.9 N/m^, 24 hours and 40 hours respectively (see
Fig. 3.22). The bed shear stress during resuspension ranged from 0.14
to 0.52 N/M'^. The bulk density and shear strength profiles for each
salt concentration are shown in Figs. 3.13 and 3.14.
Rates of resuspension, £, were calculated from the concentration
time profiles in the following manner. Smooth curves were drawn through
the data points and values of the concentration were read off these
curves at 0, 5, 15, 30 and 60 minutes after each change in the bed shear
stress. Values of dC/dt were determined using a backward difference
differentiating scheme. Values of e were calculated using Eq. 3.12.
The logarithm of the erosion rate was plotted against the average
normalized excess shear stress, i.e. {^^  where is the
average shear strength of the bed layer that was eroded by the bed shear
stress 1^. Figure 3.28 shows these plots for the 1 ppt salinity test.
The slope of each line, a, and the ordinate intercept, e^, were
determined from each graph. The values of a and plotted as a
function of depth for each salt concentration are given in Figs. 3.29
and 3.30 respectively.
Before evaluating these results it is appropriate to discuss
parameters other than the salt concentration that varied from test to
test, in order to examine the possible significance of their variance on
the rate of erosion. The other uncontrolled parameters were the
temperature, pH and the SAR of the eroding fluid. Also, the rotating
annular flume does not have the facility to maintain a constant water
97
Z Tm =0.9N/m^
 Tm =24 hrs.
0 0.05 0.1 0.15 0.2
(^)
Fig. 3.28. Resuspension Rate Versus Normalized Excess Shear Stress,
98
£
0 2 4 6 8 10
. 3.29. Slope, a, Versus Depth Below Bed Surface, z, , as a
Function of Salinity. °
3.30. Ordinate Intercept, e^. Versus Depth Below Bed Surface,
Zj^, as a Function of Salinity.
99
temperature during the course of an experiment. As a result the
temperature typically varied 3° to 5°C over the seven to eight hour
duration. A temperature variation of this magnitude has been found to
result in less than a 3% decrease in the bed shear strength (Kelly and
Gularte, 1981) and is considered to be insignificant. Likewise, over
the one percent salt concentration range used in these experiments the
variation in pH is considered to be not significant. However, due to
the relatively small quantities of Ca^"*" and Mg^"*" compared to that of
Na"*", the SAR values were rather large and increased significantly with
increasing salt concentration. For example, the SAR values varied from
110 to 349 as the salinity increased from 1 to 10 ppt. Alizadeh (1974)
showed that both the concentration of the electrolyte and the SAR are
important controlling factors in the process of coagulation.
Specifically he found that the effect of salt concentration gradually
decreases with increasing SAR. Thus, the varying SAR values are
considered to have had some, albeit unmeasured, effect on these
experiments.
Analysis of the variation in the bed density profiles with
salinity, shown in Fig. 3.13, revealed no discernable relationship. It
is felt that further investigations are necessary to determine if any
relationship exists between Pg and the salt concentration of the eroding
fluid for a stratified cohesive sediment bed.
The bed shear strength profiles, shown in Fig. 3.14, were analyzed
by determining the weighted depthaveraged value (weighted with respect
to spacing, i.e. depth, between adjacent data points) of at the five
different salt concentrations, S. The following relationship was found:
i
100
^^(S) = T^(S=0)*(S/2 + 1) for 0 £ S < 2
(3.14)
^^(3) = 2t^(S=0) f or S >_ 2
where S is in ppt. This relationship was incorporated into the bed
formation algorithm in the following manner. The discretized value of
at the top of the uppermost new deposit bed layer is changed
instantaneously, i.e. during the same timestep, at every element where
the elemental average salinity value changes. For the second bed layer
the discretized value is not changed during the first timestep
during which the average value of salinity changes; it is changed at the
timestep during which the salinity changes for the second time.
However, for this bed layer the new value is determined using Eq.
3.14 and the second preceding value of the salinity at that element.
This procedure is similarly repeated for the remaining new deposit bed
layers. This method of incorporating the effect of the salinity of the
eroding fluid on the bed shear strength profile was used in order to
account, at least partially, for the finite amount of time it takes for
denser (i.,e. higher salinity) eroding fluid to diffuse downward into the
bed or for denser pore fluid to diffuse upward into the overlying
eroding fluid. The diffusion coefficients of CI" and Na"^ in Pacific red
clay and Lake Ontario sediment were experimentally determined to be of
the order of 10"^  10"^ cm^/sec at a temperature of 24°C (Li and
Gregory, 1974; Lerman and Weiler, 1970). These extremely small
diffusion coefficients indicate that the rates of diffusion in
unconsolidated sediments are generally from one half to one twentieth of
the diffusion rates in the eroding fluid (Manheim, 1970).
101
For the first timestep the initial salinity value at each element
is used to determine the values in both the unconsolidated and the
partially consolidated bed layers, while the salinity of the pore water
in the original settled bed layers, an input parameter in the model, is
used to evaluate the values in this bed section. The i^ values of
the settled bed layers are thereafter assumed to be invariant with
respect to the salinity of the eroding fluid. The justification for
this assumption is based on the observation that dissolved silica
concentrations in pore waters of Lake Ontario, Erie and Superior
sediments were, in general, invariant with respect to depth after the
first 20 cm below the watermud interface (Nriagu, 1978). Therefore,
the salinity of the eroding fluid would not be expected to influence
that of the pore fluid below the top 20 cm of the bed, which clearly
encompasses the consolidated bed section.
The values of a and are seen in Figs. 3.29 and 3.30, to decrease
and increase, respectively, with increasing salinity. However, inasmuch
as these parameters are considered to be characteristic properties of
the sediment bed, and as the effect of salinity on another bed property,
c^, which is as well estimated indirectly from measured data, has
already been incorporated into the model, it was not necessary to
consider the variation of a and with salinity in the erosion
algorithm.
3.4.3. Erosion Algorithm
A description of the redispersion and resuspension algorithms is
given below. In both algorithms, the rate of erosion is calculated on
an element by element basis.
102
A portion of the unconsolidated new deposit (UND), when present,
will redisperse (mass erode) when t:^ is greater than the surface shear
strength of the UNO, i.e. ^^{2^=0). The thickness of the UND that fails
totally and is instantly redispersed is equal to z^^, where z^^ = bed
depth at which ■^^(25) = "^5 The value of z^^^ is determined from the
linearly varying '^f.iz^) profile in each UND layer. The value of z^^ may
be greater than the thickness of the top layer, TLAYM(l), in which case
more than one layer is redispersed. The dry mass of sediment that is
redispersed, Mp, is calculated according to
= J P(z )dz (3.15)
0
where Mq has units of Kg/m'^ and is considered to be the mass eroded over
one timestep At. The contribution to the source term in the governing
equation (Eq. 3.5) caused by redispersion is given by Eq. 3.15 divided
by the product of the average elemental water depth and the timestep
At. New UND layer(s) thicknesses and '^f^iz^) and p{z^) profiles are
calculated at each timestep when redispersion occurs by subtracting z^
and resetting T:^iz^=0) and p(Z^=0) equal to the respective initial
values at z^ = Zj^^. If z^^ is calculated to be greater than the
thickness of the entire UND, then all of this sediment is redispersed.
For both the redispersion and resuspension algorithms, erosion is
considered to occur only during accelerating flows, i.e. Tj^(t+At) >
\it). Thus, even though ^{^(t+At) may be greater than x^{z^=0), no
erosion will occur if > ^^(t+At). This stipulation for the
occurrence of erosion, and an analogous one for deposition (as will be
discussed in Section 3.6), is based on an interpretation of the
103
typically observed Eulerian timeconcentration variation in an estuarial
environment. For example. Fig. 1.3 shows a timeconcentration profile
from the Savannah River estuary (Krone, 1972). Also indicated is the
observed correlation between accelerating flows and increasing
suspension concentration and between decelerating flows and decreasing
suspension concentration. Laboratory evidence (Mehta and Partheniades,
1975; Partheniades, 1977; Mehta et^., 1982a; Parchure, 1983) suggests
that under accelerating flows, erosion occurs without redeposition of
the eroded sediment. Likewise, during decelerating flows, sediment
deposits without reentrainment of the deposit. During periods of steady
flows, erosion or deposition may occur. These two processes do not,
however, occur simultaneously even in this case (Parchure, 1983). The
initial condition at the inception of the steady flow period will
determine whether erosion or deposition will occur. If the antecedent
phase was one of acceleration, the sediment will continue to erode under
the steady flow condition. In both cases, however, relatively short
transient periods of simultaneous erosion and deposition sometimes do
tend to occur (Yeh, 1979). For estuarial modeling purposes, however,
these periods may be ignored without introducing any significant errors.
Resuspension of partially consolidated beds (CND) occurs when: 1)
the entire UND has been redispersed, 2) T^(t+At) > "^^it) and 3) 'i:^(t+At)
> ^^(21^=0), where z^=0 is now at the fluidCMD interface. The
resuspension rate expression (Eq. 3.13) found by Mehta _et al_. (1982a) is
used to determine the thickness of the CND, zu , that is resuspended
during a timestep. At. The following iterative procedure is used to
calculate z^^ during any given timestep.
104
The average erosion rate, e, for the period At is calculated as:
I =1/2 [£(t) + e(t+At)] (3.16)
in which
(t+At)
D
e(t+At) = e^CDexpr (1)( 1)] (3.17)
c
where ^qH) and a(i) are the average empirical coefficients for the
first (i.e. top) CMD layer, and
1 2.
'\ =^^2!:\(Zb=0) + — / ^*^^(Zj^)dz] (3.18)
As a first guess, z^^ is set equal to TLAY(l) = Zj^^^ (see Fig. 3.15).
A new value of Zj^ , designated Zj^ , is calculated according to:
* *2
(3.19)
where p is the average dry bed density over the first bed depth Z[^^_^.
Then the following parameter is evaluated:
^  1 = ixil (3.20)
A\ = . 1 = \xl\
sAt
105
where p and e are determined using z = z . If aa. < 0.02, then Zu is
taken to be the depth of bed eroded during this timestep. If AA. >
0.02, then yet another new value of z^^^, designated is calculated
using the following equation:
Z =z + 1 (3.21)
"*3 \,  \
where = P'zu /(e*At^' '^''"'^^ ^ ^""^ ^ determined using Zj, = z^ .
^^■'"9 ^b^vo' ^'^^ entire procedure, i.e. Eqs. 3.16 through 3.21, is
*3
repeated until the chosen error criterion, i.e. aa < 0.02, is
satisfied. As in the redispersion routine, new CND layer thickness(es)
and and p(z^) profiles are determined. As before, Z^ may be
greater than the thickness on the top layer. Laboratory tests required
to evaluate i^c^^b^' *^^^b^» ^""^ average values of and a for each
CMD layer are described in Appendix D, Section D.2.
Once the entire new deposit bed section has been eroded, the
original settled bed, if any exists, will undergo resuspension when the
following two conditions occur: 1) t^(t+At) > \{X.) and 2) T[^(t+At) >
'Cj,(Zfj=0), where Zfj=0 is now at the top of the settled bed. The surface
erosion rate expression (Eq. 3.12) given by Ariathurai and Arulanandan
(1978) is used to evaluate the thickness, z^,^, of the settled bed that
is eroded during each timestep. The iterative procedure used for the
CND is again used to solve for Zj,^, with only the expression for e being
different,. Equation 3.16 becomes
106
e(t+At) = M(l)( : 1) (3.22)
c
where M(l) is the erodibility constant for the first layer.
The contribution to the source term in Eq. 3.5 caused by
resuspension is given by Eq. 3.16, with Eq. 3.17 used for the partially
consolidated bed section and Eq. 3.22 used for the original settled bed
section, divided by the average elemental water depth.
In the following section, the dispersive transport of suspended
sediments is discussed, followed by a description of the dispersion
algorithm.
3.5. Dispersive Transport
3.5.1. Dispersion Mechanisms
There have been numerous studies on the dispersion of some quantity
(e.g. sediment) in a bounded shear flow in the thirty years since the
work of Taylor (1953, 1954). Taylor proved that a onedimensional
dispersion equation can be used to represent the longitudinal dispersion
of a quantity in turbulent pipe flow. Taylor's analysis has since been
extended to shear flow in both rivers and estuaries. The present
discussion is limited to dispersion in estuary flow. A brief review of
dispersive transport theory precedes that of estuarial dispersion.
The governing equation (Eq. 3.5) derived for the twodimensional,
depthaveraged movement of suspended sediment in a turbulent flow field
includes dispersive transport terms which account for the transport of
sediment by processes other than advective transport. Some of these
other processes include the effects of spatial (i.e. transverse and
vertical) velocity variations in bounded shear flows and turbulent
107
diffusion. Thus, the effective sediment dispersion coefficients in Eq.
3.5 must include the effect of all processes whose scale is less than
the grid size of the model or what has been averaged over time and/or
space (Fischer et__al_. , 1979). For example, the effect of the actual
vertical concentration gradient would have to be incorporated in the
dispersion coefficients in the present depthaveraged transport model.
Diffusion is defined as "the transport in a given direction at a
point in the flow due to the difference between the true advection in
that direction and the time average of the advection in that direction,"
and dispersion is defined as "the transport in a given direction due to
the difference between the true advection in that direction and the
spatial average of the advection in that direction" (Holley, 1969).
Holley enunciates the fact that diffusion and dispersion are both
actually advective transport mechanisms, and that in a given flow field,
the relative importance of one mechanism over the other depends on the
magnitude of the concentration gradient in the particular transport
problem. In the governing equation (Eq. 3.5) the effective sediment
dispersion coefficients are equal to the sum of the turbulent diffusion
coefficients and dispersion coefficients. This approach follows the
analysis of Aris (1956) which showed that the coefficients due to
turbulent diffusion and shear flow (dispersion) were additive. Thus,
the analytic expressions to be used for the effective sediment
dispersion tensor would include, at least in some sense, both diffusion
and dispersion.
Fischer (1966) showed that the dispersion of a given quantity of
tracer injected into a natural stream is divided into two separate
phases. The first is the convective period in which the tracer mixes
108
vertically, laterally and longitudinally until it is completely
distributed across the stream. The second phase is the diffusive period
during which the lateral and possibly the vertical (depending on the
nature of the tracer) concentration gradient is small, and the
longitudinal concentration profile is highly skewed. The governing
equation (Eq. 3.5) is strictly valid only in the diffusive period. The
criterion for determining if the dispersing tracer is in the diffusive
period is if it has been in the flow longer than the Lagrangian time
scale and has spread over a wider distance than the Lagrangian length
scale (Fischer et_ al_., 1979). The latter scale is a measure of the
distance a particle travels before it forgets its initial conditions
(i.e. position and velocity).
Analytic expressions for the sediment (mass) diffusion coefficients
can be obtained by analogy with the kinematic eddy viscosity.
Specifically, the Reynolds analogy assumes that the processes of
momentum and mass transfer are similar, and that the turbulent diffusion
coefficient, E, and the kinematic eddy viscosity, e^, are in fact
linearly proportional. Jobson and Sayre (1970) verified the Reynolds
analogy for sediment particles in the Stokes range (less than about 100
m in diameter). They found that the "portion of the turbulent mass
transfer coefficient for sediment particles which is directly
attributable to tangential components of turbulent velocity
fluctuations: (a) is approximately proportional to the momentum
transfer coefficient and the proportionality constant is less than or
equal to 1; and (b) decreases with increasing particle size."
Therefore, the effective sediment mass dispersion coefficients for
cohesive sediments may be justifiably assumed to be equal to those for
the flow itself.
109
Fischer ^ a]_. (1979) define four primary mechanisms of dispersion
in estuaries: 1) gravitational circulation, 2) shearflow dispersion,
3) bathymetry induced dispersion and 4) windinduced circulations.
These four mechanisms are briefly described next.
Gravitational or baroclinic circulation in estuaries is the flow
induced by the density difference between the fresh water at the
landward end of the estuary and the sea water at the ocean end. There
are two types of gravitational circulation. Transverse gravitational
circulation is depthaveraged flow that is predominantly seaward in the
shallow regions of a crosssection and landward in the deeper parts.
Figure 3.31a depicts this net depthaveraged upstream (landward) and
downstream (seaward) transport and the resulting transverse flow from
the deeper to the shallower parts of the crosssection. Fischer et al .
(1979) state that "the upstream flow is expected to be concentrated in
the deeper portions of the channel, because the upstream pressure
gradient increases linearly with depth below the water surface." Thus,
the interaction between the crosssectional bathymetry and the
baroclinic flow causes the transverse circulation.
Vertical gravitational circulation is schematically illustrated in
Fig. 3.31b which shows the predominantly seaward flow in the upper part
of the flow and landward flow in the lower part. Fischer (1972)
believes that the vertical gravitational circulation will be more
important than transverse circulation only in highly stratified
estuaries.
The previously described mechanism of shearflow dispersion is
believed to be the dominant mechanism in long, fairly uniform sections
of wellmixed and partially stratified estuaries (Fischer et al..
110
(a)
NET DOWNSTREAM TRANSPORT
"NEAR THE SURFACE
VECTICAL AOVECTION
AND DIFFUSION
Fig. 3.31,
NET UPSTREAM TRANSPORT
NEAR THE BOTTOM
(b)
The Internal Circulation Driven by the River Discharge in
a Partially Stratified Estuary, (a) A Transverse Section;
(b) A Vertical Section (after Fischer et al_. , 1979).
Ill
1979). Holley et aT_. (1970) applied the dispersion analysis of Taylor
(1954) to oscillating flow in estuaries. They concluded that for wide
estuaries, the effect of the vertical velocity distribution on shear
flow dispersion is dominant over that of the transverse velocity
distribution. The exact opposite situation was found for relatively
narrow estuaries.
The joint influence of bathymetry and density differences on
dispersion has already been mentioned in the discussion on baroclinic
circulation. Other examples of bathymetry induced dispersion include:
the intrusion of salinity or sediment in certain parts of a cross
section caused by the channelization of flood and ebb tides in tidal
inlets or narrow estuaries (Fischer etal_., 1979); and the enhanced
dispersion of a quantity (e.g. pollutant) or intrusion of salinity in
tidal flats and side embayments, which serve as storage areas for these
constituents, caused by the out of phase flow which occurs between the
main channel and such features (Okubo, 1973).
An example of a windinduced circulation is shown in Fig. 3.32.
The steady onshore wind causes a circulation in the wind direction in
the shallow bay, where the less water mass per unit surface area results
in a higher acceleration and therefore quicker response to the wind
induced surface stress, and in the opposite direction in the deeper
sections of the channel. Such a circulation can cause significant
dispersion (Fischer _et_al_. , 1979).
For a detailed description of dispersion, the reader is referred to
the following references: Dispersion in estuaries  Glenne and Selleck
(1969), Pritchard (1969), Holley (1969), Holley etal_. (1970), Fischer
(1972), Okubo (1973), Ward (1976), Fischer (1978), Murray and Siripong
112
(1978), Zimmerman (1978) and Fischer ^al_. (1979); Dispersion in
channel flows  Fischer (1966), Dagan (1969), Fischer (1970), Peterson
et al_. (1974), Taylor (1974), Ward (1974), Sumer and Fischer (1977),
Smith (1978), Beltaos (1980a, b), Chatwin (1980) and Liu and Cheng
(1980); Dispersion of particle matter  Sayre (1969), Jobson and Sayre
(1970), Chen (1971), Sumer (1971) and Alonso (1981).
3.5.2. Dispersion Algorithm
The most important, and possibly the most difficult task in
modeling dispersion is to determine which of the dispersion mechanisms
are important in the estuary being modeled. For example, if the estuary
has only a few tidal flats and shore irregularities and has a fairly
uniform crosssection (e.g. the Delaware River), shear flow dispersion
may be the dominant mechanism. However, if the estuary is relatively
deep and the river discharge is large (e.g. the Mississippi River),
gravitational circulation may be just as or even more important than
shear flow dispersion. Unfortunately, none of the existing dispersion
models, most of which are twodimensional (e.g. DISPER (Leimkuhler _et_
al . , 1975)), can represent the combined effects of, for example, an
irregular shoreline configuration and bathymetry, shear flow dispersion
and baroclinic flow.
Because of these problems in identifying, describing and modeling
the various dispersion mechanisms which occur in estuaries, the decision
was made to develop a dispersion algorithm for only shear flow
dispersion that would be applicable to a wide, vertically well mixed
estuary. Following the analysis of Hoi ley et al . (1970), it is assumed
that the dispersion in wide estuaries is associated primarily with the
113
vertical shear. The limitations, which determine the applicability of
such a dispersion algorithm, are consistent with those associated with a
twodimensional, depthaveraged cohesive sediment transport model.
The dispersion tensor derived by Fischer (1978) for two
dimensional, depthaveraged bounded shear flow is used in the dispersion
algorithm. The four components of this tensor are
D^y = (UVd2/E)I^y (3.23)
Dyj^ = (UVd2/E)Iy^
Dyy = (V2d2/E)lyy
I
in which: U and V are the rootmeansquare values of u
and V over the depth d;
u' = u(z)  u, where u is the depthaveraged
component of the velocity in the x
di recti on;
V = v(z)  V, where v is the depthaveraged
component of the velocity in the y
di recti on;
E = mean value of the scalar turbulent
diffusion coefficient in the vertical
direction, E^; and
1 „ CI C „
I. . = Ju. / — Ju.dCdCdC (3.24)
IJ 0 % E' 0 ^
114
in which: E = E^/E, u^ = u^/u and C = z/d. The quantities u and v
are the velocity deviations taken over the depth from the respective
depthaveraged values, u and v. The values U and V represent the
I I
"intensity" of u and v , respectively (Fischer et_al_., 1979). The
physical interpretation of the cross product dispersion coefficients D^^y
and Dy^ is that a velocity gradient in the direction can produce mass
(dispersive) transport in the Xj direction.
Fischer (1978) notes that since in most investigations the vertical
velocity profile, i.e. u(z) and v(z), and the vertical turbulent
diffusion coefficient, E^, are not known with a high degree of accuracy,
it would usually suffice to assume that the value of I^j in Eq. 3.24 is
a constant. The value of I^j in various parallel shear flows ranges
from 0.054 for turbulent pipe flow to 0.10 for laminar flow with a
linear velocity profile over d (Fischer et_al_. , 1979). Therefore,
Fischer recommends that a value of 0.10 be used for I^j in Eq. 3.23.
The following expression for E^, derived by Elder (1959) for flow
down an infinitely wide inclined plane, is used in this analysis:
E = Ku.zd— ) (3.25)
^ ^ d
where < = von Karman turbulence constant, and u^ = shear velocity.
Therefore, E is given by
— Id <u d
E =  / E dz = — — = 0.067u^d
d 0 ^ 6 ^
(3.26)
115
with K = 0.40. The values of < obtained by Gust (1976) from the slopes
of measured clay suspension velocity profiles varied between 0.3 and
0.4. Gust considered this variation a result of the mean flow
experimental error of S% and not due to the presence of suspended
cohesive sediments. Therefore he assumed < = 0.40 in his analysis.
This is the justification for using this value of ^ in this study.
Fischer (1966) found that in both laboratory experiments and in
real streams that the mean value of U^/u^ was equal to 0.2.
Substituting this value, and I^ ■ = 0.1 and Eq. 3.26 into Eq. 3.23 gives
XX
0.2(u)^d^ (u)^d
D = (0.10) = 0.30
n.067u^d u^
0.2uvd^ uvd
D = (0.10) = 0.30
0.067u^d
f "f
0.2vud ^ uvd
(0.10) = 0.30
(3.27)
0.067u^d u^
0.2(v)^d^ (v)^d
D = (0.10) = 0.30
yy
0.067u^d u^
These are the coefficients used in the dispersion algorithm to model the
shear flow dispersion of suspended cohesive sediments in a wide, well
mixed estuary. Values of D^.j are calculated at each time step in the
model using the specified nodal values of u, v and d.
116
In the following section, the depositional behavior of cohesive
sediments is summarized, followed by a description of the deposition
algorithm,
3.6. Deposition
3.6.1. Previous Investigations
Deposition has been defined to occur when t:^ is not high enough to
resuspend sediment material that settles onto and bonds with the bed
surface. This process, therefore, involves two other processes,
settling and bonding, i.e. cohesion.' Laboratory studies on the
depositional behavior of cohesive sediment in steady turbulent flows
have been conducted by, among others. Krone (1962), Rosillon and
Volkenborn (1964), Partheniades (1965), Partheniades et al_. (1966),
Migniot (1968), Lee (1974), Mehta and Partheniades (1975) and Mehta et
al . (1982b). The results from these and other studies on the settling
rates of cohesive sediments pertinent to the deposition algorithm
described in Section 3.6.4 are summarized below.
In laboratory flumes, the depositional behavior is usually
investigated by allowing sediment suspended in a flume at a high shear
stress to deposit by reducing the shear stress. Since the sediment
concentration gradient in the direction of flow in usually small, the
observed timerate of change of the depthaveraged concentration, C, is
due to the deposition of suspended material. The conservation of
sediment mass can be expressed as (Einstein and Krone, 1962):
.r (OC
dC d s
dt
(3.28)
117
where t = time, d = flow depth, WgCO = sediment settling velocity as a
function of C, and = probability of deposition, or the probability of
a sediment particle or floe bonding to the bed and not being instantly
resuspended.
Krone (1962) conducted a series of depositional tests in a 31 m
long and 0.90 m wide recirculating flume using mud from San Francisco
Bay which contained approximately equal proportions of clay and silt.
Krone postulated that P^ increases linearly with a decrease in
according to
P . = 1 (3.29)
cd
where = critical shear stress for deposition, above which no
deposition occurs. Therefore, P^ decreases linearly from a value of 1.0
at T^f, = 0, to 0 for > x^^. The value of was found to be equal to
0.06 N/m^ for the Bay mud with C < 0.3 g/1. Krone found that when C <
0.3 g/1, was independent of C. In this case, integration of Eq. 3.29
gives
C d s
— = exp[ 1] (3.30)
C„ d
0
where Cq is the initial suspended sediment concentration. Thus,
according to this equation all the suspended sediment will eventually
deposit when '^b < t j.
118
For 0,3 g/1 < C < 10 g/1 and for C > 10 g/1 , logarithmic laws of
the following form were derived:
log C = K[log(t)] + Constant (3.31)
where K was found to be a function of d and P^^. Krone attributed the
variation of the depositional properties with suspension concentration
to different forms of settling. Various forms of settling of coagulated
cohesive sediments are discussed later in this section.
Partheniades (1965) conducted deposition tests in a open, flow
recirculating flume using Bay mud. He noted that for flows above a
certain critical bed shear, the suspended sediment concentration, after
an initial period of rapid deposition, approached a constant value,
which he referred to as an equilibrium concentration, Cg^. The ratio
C^„/C = C* was found to be a constant for given flow conditions,
regardless of the value of Cg. Whereas for bed shears even slightly
less than this critical value, all the sediment eventually deposited.
Partheniades et_ al_. (1966) conducted deposition experiments in a
rotating annular flume (similar to the one at the University of Florida,
but with mean diameter of 0.82 m and 0.19 m wide) at the Massachusetts
Institute of Technology using a commercial grade kaolinite. Based upon
these experiments it was concluded that Cg^ represents the amount of
sediment that, having settled to the near bed region, cannot withstand
the high shear stresses present there (due to insufficiently strong
interparticle bonds) and are broken up and resuspended. In addition,
Cgq in the fine sediment deposition tests appears not to be the result
of an interchange between suspended and bed material as it is for
119
cohesionless sediment, because if such were the case, Cg^ would not be
dependent on C^. Therefore, it follows that C^^ does not represent the
maximum sediment carrying capacity of the flow, as it does in the case
of cohesionless sediment, but instead may be considered to be the steady
state concentration (Mehta and Partheniades, 1973).
As noted by Mehta and Partheniades (1975), Krone did not observe
Cgq in his tests because most of them were conducted at <
wherein C would be expected to be equal to zero. It is apparent that
the definition of ?^ must be extended to include bed shear stresses
greater than t^^.
Mehta and Partheniades (1975) investigated the depositional
properties of a commercial grade kaolinite in distilled water and in
salt water at seawater salinity (35 ppt) in a rotating annular flume
facility at the University of Florida. Figure 3.33 shows typical
suspended sediment concentrationtime plots found in these tests. It is
evident that a steady state concentration was reached in each test and
that for bed shears above approximately 0.16 N/m^, the value of Cgq was
greater than zero and in fact increased monotonical ly with increasing t:^.
Figure 3.34 shows the ratio C*q = Cgq/Cg plotted against for all
the tests with kaolinite in distilled water. Two important conclusions
are obtained from this figure: 1) C*q is a constant for a given
(and type of sediment) and is not a function of depth, d, or C^, and
2) for "^s < . C* = 0. The first conclusion is based on the
° "mi n ^"
observation that the data points for all the different flow conditions
are almost randomly scattered about a "best fit" line. The minimum bed
shear tu > observed in Fig. 3.34 is the same as the \d value defined
by Krone (1962), and the critical bed shear obtained by Partheniades
120
Cr^ — ^
ojevjojco'*?
^odcidcidcio
X d — o> irt (Ti (o d't
oOOOOOO
•ddbddddo
0)
CD
n3
00
Q
C
+>
o
S
o
4
0)
(/I
CU
o o
o
+J
fa
cn
s
cn
121
(1965). As observed in this figure, found to be approximately
'^min
0.18 N/m^ for kaolinite in distilled water.
In Fig. 3.35 the data of Fig. 3.34 are plotted on lognormal
coordinates as in percent against where \ = ^b'^'^b^^^*
straight line through the data points gives the following relationship
between these two dimensionless parameters:
C =1/2 (1 + erf (— ))
eq
/2
(3.32)
with
^a = ^'ho^
(3.33)
where is the standard deviation and (v^^50 geometric mean of
the lognormal relationship given by Eq. 3.32, and erf is the error
function (Mehta and Partheniades, 1975).
It was found that for all tests with kaolinite in both distilled
and salt water, with a 50/50 mixture of kaolinite and San Francisco Bay
mud in salt water, with only Bay mud in salt water, and for the
reanalyzed deposition tests of Rosillon and Volkenborn (1964),
Partheniades (1965) and Partheniades et__al_. (1968), the value of was
0.49. Therefore, as noted from Eq. 3.33, C*^ is dependent solely on the
value of the ratio (Vl)/{V^^50 '^^^^^ ^"^ Partheniades ( 1973) found
for deposition tests in salt water the following relationship between
Xin ^""^ ^Vl^SO:
122
123
= 4 exp(1.27 _ ) (3.34)
mi n
Mehta and Partheniades (1975) found the following dimensionless
lognormal relationship for the variation, i.e. decrease, of the
suspended sediment concentration v/ith time:
^ T
C =1/2 (1 + erf( ) (3.35)
/2
where
T = logjQ(t/t5g) ^ (3.36)
and where: C* = (Co'C)/(CoCgq) represents the fraction of the
depositable sediment, C^Cgq, deposited at any given time t, cr^ is the
standard deviation of the lognormal relationship, and tgg is the
geometric mean (i.e. the time at which C* = 50%). Figure 3.36 shows a
comparison between some typical depositional data for kaolinite in
distilled water and the lognormal relationship given by Eq. 3.35. This
relationship was found to hold for all values of greater than
approximately 0.25, with the exception that for very high Cg values
(around 2025 g/1 ) with i;^ < 1, an acceptable agreement with the
measured data was not obtained. Good agreement was as well obtained
between Eq. 3.35 and the data sets mentioned previously in this section.
Taking the derivative of Eq. 3.35 with respect to time gives the
following expression for the rate of change of C* y^ith time:
124
125
dC* 0.434 exp(T^/2)
= (3.37)
dt /2i t
The standard deviation, and the geometric mean, t^g, were found
to be functions of X]^, d, and C^. Shown in Figs. 3.37 and 3.38 are
examples of the relationships found between these parameters. The
following conclusions were arrived at from these and other similar plots
given by Mehta (1973): 1) for a specific value of t^, the deposition
rate was minimum. This tj^ value was found to vary between 1 and 2 for
kaolinite in distilled water. The rate of deposition increased for
values both less than and higher than this specific value, but not as
significantly for higher values as for the lower values. However, for
^b » deposition of suspended sediment occurred. For Bay mud
max
in sea water, Xu was determined to be 1.69 N/m'^. 2) for x. < 1, the
■^max
rate of deposition increased with an increase in d, while for Xj^ > i,
the effect of d on the deposition rate was minimal.
As noted, the settling velocity of suspended cohesive sediment
particles has been found to be a function of, among other parameters,
the suspension concentration (Krone, 1962). There appears to be at
least three types of settling: 1) no mutual interference, 2) mutual
interference and 3) hindered settling. For very low suspension
concentrations, on the order of 0.10.7 g/1 , the aggregates or
elementary particles settle independently without much mutual
interference, and therefore the settling velocity is independent of C.
For concentrations between approximately 0.3 g/1 and 1015 g/1, the
settling velocity increases with concentration due to the accompanying
increase in interparticle (floe) collisions, and therefore increased
mutual interference (Fig. 3.39). For concentrations higher than 1015
126
Fig. 3.37. Log t^Q Versus for Kaolinite in Distilled Water (after
Mehta and Partheniades, 1975).
■k
Fig. 3.38. Versus for Kaolinite in Distilled Water (after Mehta
and Partheniades, 1975).
127
C \ I I I I I 1 I I I
0.1 0.2 0.4 0.6 0.8 1.0 2.0
SUSPENDED SEDIMENT CONCENTRATION, C(g/^)
Fig. 3.39. Settling Velocity, W^, Versus Suspended Sediment
Concentration, C, for San Francisco Bay Mud (after
Krone, 1962).
128
g/1, the settling velocity actually decreases with increasing
concentration (Figs. 3.40 and 3.41). At such high concentrations the
sediment suspension, referred to as fluid mud or mud cake (Bellessort,
1973), hinders the upward flux of water expelled by consolidation of the
lower suspension (Krone, 1962).
In the mutual interference range. Krone (1962) and Owen (1971) have
found the following empirical relationship between the median settling
velocity, Wg, and C:
= Kc" • (3.38)
where K and n are the empirical constants that depend on the sediment
type and the turbulence intensity of the suspending fluid. Krone found
n to be equal to 1.33 for Bay mud in laboratory experiments (see Fig.
3.39). Teeter (1983) found n to be less than 1.0 for sediment from
Atchafalaya Bay, Louisiana.
Owen (1971) studied the effect of turbulence on the settling
velocities of natural mud. No description of the sediment was reported,
except that it was collected in the Thames River near Dagenham,
England. A specially designed sampling instrument was used to collect
sediment samples during both a spring tide and a neap tide. This tube
collects undisturbed samples of suspended sediment in an estuarine
environment, and immediately thereafter the median settling velocity of
"natural aggregates" can be determined using the bottom withdrawal
method described by Owen (1970). The value of n determined using this
method was 1.1 and 2.2 for sediment collected during a spring and a neap
tide, respectively. The turbulence intensity during a spring tide is
129
9 3.0
X
in
£
E
o
o
_l
>
I
I
u
C/0
SUSPENDED C0NCENTRAT10N,C(g/i)
Fig. 3.40. Settling Velocity, W^, Versus Suspended Sediment
Concentration, C, for Yangtze River Estuary Mud
(after Huang et al . , 1980).
Fig. 3.41,
i;; 100.0
£
£
>
I
UJ
>
UJ
CO
10.0
.0
0.
O.Oi
<
S 0.0 i
O.IO
Severn estuary mud
(saline water)
1.0 10.0 100.0
SUSPENDED SEDIMENT CONCENTRATION,
C (g/^)
Versus C for Severn Estuary Mud (after Thorn, 1981)
130
greater than during a neap tide. Owen therefore postulated that n was
greater (and therefore as well) during the neap tide because the
lower level of turbulence did not cause significant breakage of the
aggregates; thus relatively large aggregates with higher settling
velocities were formed. During the spring tide the higher degree of
turbulence, and therefore greater internal shearing rates, did result in
breakage of a significant proportion of the aggregates. Thus, small
aggregates with lower settling rates, and therefore lower values of n,
were formed. Owen believed that the interparticle collision rate was
significantly high during both tides and therefore did not consider it
very probable that the aggregate size would have been affected (i.e.
limited) by this factor. Owen also performed standard settling tests in
a one meter high bottom withdrawal tube using the same sediment samples
as above, and found that varied linearly with C (i.e. n==1.0) for both
spring and neap tide samples and that the values of were
approximately one order of magnitude smaller than the values
determined with the aggregate collection tube. The latter result is
very significant in that it reveals the apparent effect of turbulence on
the behavior (e.g. settling velocities) of sediment aggregates: larger,
stronger aggregates with corresponding higher settling velocities are
formed in a turbulent flow field than under quiescent conditions
primarily because of increased collision rates due to high internal
shearing rates.
Migniot (1968) defined a "flocculation factor" F, given below, in
order to quantify the effect of the aggregation intensity on Wg:
131
W
F = (3.39)
W
where W is the median settling velocity of the aggregates and W is
A
the median settling velocity of the elementary sediment particles.
Bellessort (1973) reported that F varies with the grain size of the
elementary (i.e. deflocculated) particles according to
F = a^.D = (3.40)
where D is the mean diameter of the particles in microns (10~^m), oc^ =
250 and = 0.9, provided is measured in mm/s. Figure 3.42 shows
this effect of the particle size on F and W. for numerous sediment
samples at = 10 g/1 and salinity S = 30 ppt. Also plotted in this
figure is the variation of F with D found by Dixit _eta][. (1982) using
mud from Atchafalaya Bay, Louisiana. However, in these data Cg varied
from 1.2 to 11 g/1 as indicated and S=0.0 ppt. Another important
difference between the two data sets is that Bellessort measured W.
^A
under quiescent conditions, while Dixit _et__al_. (1982) measured this
settling velocity under turbulent flow in the rotating annular flume.
As observed, these data have the same slope between F and D as given by
Bellessort, This suggests that, in general, F may be proportional to
D albeit with different intercept values, at least for suspension
concentrations with C = 1.211 g/1 and 0<S<30 ppt.
132
MEAN DIAMETER (Microns)
VELOCITY (mm/s)
Fig. 3.42. Effect of Size and Settling Velocity of Elementary Particles
on the Coagulation Factor of Natural Muds after Bellessort,
1973).
133
3.6.2. Effect of Salinity on Deposition
The larger, stronger aggregates of natural muds formed in a saline
medium have been found to result in higher settling velocities (Krone,
1962; Owen, 1970), which result in higher rates of deposition. Thus,
the effect of salinity on the deposition of cohesive sediments may be
quantified in terms of a relationship between salinity and the median
settling velocity, Wg, of a particular sediment.
Krone (1962) studied the effect of salinity and suspended sediment
concentration on of sediment from Mare Island Strait in San Francisco
Bay. Hydrometer analysis showed that 60% by weight of this sediment was
in the clay size range (i.e. < 2 urn), with the remainder in the silt
size range. Xray diffraction and differential thermal analyses of the
clay fraction revealed a large content of illite, montmorillonite and
kaolinite clay groups along with small quantities of chlorite and
quartz. The results from settling tests performed under quiescent
conditions in oneliter cylinders showed the effect of both salinity and
suspension concentration on (Fig. 3.43). The influence of salinity
on is especially significant in the range 02 ppt, particularly for
the 1.0 and 0.53 g/1 suspension concentrations. This result is expected
considering the discussion presented in Section 2.2.6. One possible
explanation for the apparent increasing influence of salinity on with
increasing suspension concentration, as shown in this figure, is the
following. As the suspension concentration increases, the number of
collisions (by Brownian motion and differential settling mechanisms in
such an experiment) would likewise increase and therefore promote the
formation of larger aggregates with higher settling velocities. The
lowest order aggregate that could be formed would be limited by the
i
134
135
suspension concentration, so that even with an increase in salinity (and
therefore a corresponding increase in cohesive forces), lower order
aggregates with typically higher settling velocities could not form due
to the insufficient concentration of suspended particles.
Owen (1970) studied the variation of Wj of a natural mud with
salinity and suspension concentration. Approximately 55% of the mud was
in the clay size range, with the remainder in the silt range. It was
revealed that the clay fraction was composed of, in order of abundance,
illite, kaolinite, montmoril lonite and chlorite. Settling tests were
conducted in a two meter high bottom withdrawal settling tube.
The results of Owen's tests are shown in Figs. 3.44 and 3.45.
These figures show that, in general, as the salinity and suspension
concentrations are increased, increased cohesion and interparticle
collision result in higher coagulation rates with accompanying higher
settling velocities. This trend corroborates that found by Krone
(1962), except that no "leveling off" of above a certain salinity
value was found in these tests. The decrease in above a given
salinity and concentration, as observed in both figures, usually
represents the onset of hindered settling. The effect of salinity on
is seen to be diminished at suspension concentrations in the hindered
settling range.
Owen (1971) found a negligible effect of salinity on the settling
velocity of natural aggregates at two different locations in the Thames
River estuary. The salinities at the two sampling stations varied
between 610 ppt and 3226 ppt, respectively. Evidently, the effect of
salinity on Wj at these salinities in a turbulent flow field is much
less than that under quiescent conditions (see Fig. 3.44). This implies
136
SALINITY, S (PPT)
Fig. 3.44. Effect of Salinity on Settling Velocity of Avonmouth Mud
(after Owen, 1970).
138
that increased cohesion caused by the higher salinities is counter
balanced by the high internal shear rates which cause the aggregates to
be broken apart.
Deposition tests were conducted at the University of Florida in
order to further investigate the effect on both salinity and bed shear
stress on the settling rates of the Lake Francis mud. The salt
concentrations utilized in these tests were: 0, 1, 2, 5, 10, 20 and 35
ppt. The tests were conducted in the rotating annular flume with a
water depth of 0.31 m at the following values of Tj^: 0.0, 0.015, 0.10,
0.20 and 0.30 M/m . The initial concentration for these tests varied
between 3,.7 and 4.7 g/1. Before the start of each test, the sediment
and water were mixed for two hours at a shear stress of 0.90 N/m^. The
shear stress was then reduced to the appropriate value and the
samples of the suspended sediment were collected at 0, 1, 2, 5, 10, 15,
20, 30 and 60 minutes after the shear stress was reduced to t^^.
Subsequent samples were collected with lower frequency. Each experiment
was conducted for a period of 21 hours.
The measured suspension concentrations were plotted against time
for each experiment. Plots of C/Cg versus time for each value at a
salt concentration of 5 ppt are shown in Fig. 3.46. The steady state
concentration, Cg^, for each deposition test was determined from a curve
drawn to represent the mean variation of the concentration with time, as
seen in this figure. The following observations were made: 1) For the
two lowest values of x^, i.e. = 0.0, and 0.015 M/m^, the
concentration decreased over the duration of the experiment for all salt
concentration values, indicating that C^^ for these x^ values would
probably have been equal to zero if the experiment had been of longer
139
140
duration. 2) For = 0.05, 0.1. 0.2 and 0.3 N/m^ the concentration
decreased rapidly during the first hour and C^q was reached more rapidly
as the value of ■^^ increased. 3) At the four lowest values of t;^ the
effect of salt concentration on the deposition rate (i.e. concentration
variation with time) was appreciable. For the two highest values the
salt effect was much less discernable. These results seem to indicate
that at relatively low values of cohesive forces are predominant,
whereas at the higher values the hydrodynamic forces (i.e. disruptive
internal shears) become at least as significant. This explanation
follows from the results obtained by Owen (1970; 1971) and by Mehta and
Partheniades (1975).
The ratio Cgq/C^ was plotted against values for all salt
concentrations (Fig. 3.47). Interpolation of the resulting plot yields
' ^""^ believed that even though additional data
(i.e. Cgq/Cg against i^ values) might have resulted in a different value
of "^b . ' values would be reasonable close (probably within ±
mi n r J
25%). Analysis of the results from these experiments is given in the
next section.
3.6.3. Deposition Rates
The product P^.W^ in Eq. 3.28 defines an effective settling
velocity, W^, which, in general, is smaller in magnitude than since
the range of P^ is between 0 and 1. The rate of deposition given by Eq.
3.28 may therefore be written as
dC s
— = (3.41)
dt d
141
Range I C
Hindered
Settling j
RangelB I
Mutual I
Interference
Range lA
No Mutual
Interference
.Not a straigtit line in general
because T^^ = f (C)
Range E B
Mutual Interference
Range H A
No Mutuallnterference
0.25 1.0
•I
b b bnnin
b max
3.48. Apparent Settling Velocity Description in Domains
Defined by Suspended Sediment Concentration and
Bed Shear Stress.
142
where is hereafter defined as the effective mean settling velocity
for a given sediment. For the dimensionless bed shear stress less
* * *
than a certain characteristic value, Xj^ ^, with the range 0 ^\<'^^
designated as Range I, and for the concentration range C<C^ for all
values of (Fig. 3.48) the following empirical relationships for
are assumed:
Pd^,l
for C < C
1
(3.42)
s ,1
P .KC
d
for < C <
(3.43)
( ) w (_ _ for c > C,
250
(3.44)
IBv
where W^j = median sediment settling velocity in the free settling
range, gD^(P3/p^l)/(18v) = and is defined by Eq. 3.29.
Therefore, depending upon the value of C, the rate of deposition in
Range I (see Fig. 3.48) is given by Eqs. 3.423.44. These three
expressions for W^j are based upon the experimental results of Krone
(1962), Ov^en (1971) and Bellessort (1973). Typical values for and €3
are 0.10„7 g/1 and 1015 g/1 respectively. The value of 1 was found to
be approximately 0.6 using the settling velocities measured by Owen
(1970), Huang etal_. (1980) and Thorn (1981). The values W. , K, n and
s , i
Ci can vary widely, depending upon, among other factors, the particle
diameter, D, the type of sediment and the salinity. These parameters
must be determined in laboratory settling tests (further discussion of
this aspect is given in Appendix D, Section D.2).
143
^max ~ ^max Kin> ^""^ concentration range C
> C^, designated as Range IIB, the rate of deposition is determined
using a lognormal relationship (Mehta and Partheniades, 1975):
dC 0.434 exp(T^/2)
dt 2y7^a^ t ^°
*
2.04 ^V^^
(lerf( log.^C ])) {3.45)
/2 ^«>^P(l27^bmin)
where = 0.49. Eqs. 3.323.34 and C* = iC^Cf iC^C^^) have been
substituted into Eq. 3.37. The following expression for in Range IIB
was determined by equating Eqs. 3.41 and 3.45:
0
k^d exp(T^/2) C
=
o^t C
2.04 (V^^
(lerf(_log^ [ ])) (3.46)
/2 4exp(1.27x. . )
bmin
*
where l<2 = 0.434/ (2/2Tt). For x^<l, the argument of the error function
is set equal to zero. This expression for {and therefore Eq. 3.45 as
well) is assumed to be valid for C>Ci for the following reason: the
phenomenon of hindered settling was not observed in the steadystate
deposition tests under turbulent flows performed by Mehta (1973) for
concentrations up to about 20 g/1 . Evidently, the higher t* values that
Mehta used in his tests prevented the occurence of this mode of
settling, inasmuch as Krone (1962) did observe hindered settling in his
tests which, in general, were conducted at lower values of cb
144
*
Deposition tests with 0.25 < t:^ < 1 using San Francisco Bay mud in
sea water and kaolinite in distilled water revealed that for suspension
concentrations less than Cj^ ==0.10. 7 g/1 , the exponential law given by
Eq. 3.41 was valid. Therefore, for C < C^^ in Range IIA, the rate of
deposition is given by Eqs. 3.41 and 3.42 with obtained from:
p = for 1 < 1 < x (3.47)
d ,, b,c b b
VJ 1 max
s,l
where W^^j is given by Eq. 3.46 with Cg=C=Cj^. Thus, for C<C^ in
Range II is defined such that the value of ii(Cj) in Range II is
equal to i=Pcl*^''s,l C<C^. Therefore, and dC/dt are continuous
functions for all concentrations in Range II.
Likewise, the parameter t:^ ^ is defined to be the value of t:^ at
which the expression for ^^(C) in Range I is equal to the same in Range
II. Thus, Wg and therefore dC/dt are continuous functions for the
* ★
entire deposition range (t;^ < 1). It is apparent that Tu is not a
max
constant, as it is a function of the depthaveraged concentration, C.
*
Solving for ^ gives
^s II
^b,c = l^ forC>C^ (3.48)
*
where Wgjj is given by Eq. 3.46, and W^^j by either Eq. 3.43 or 3.44
(depending on the value of C) divided by P^.
145
The previously described deposition tests performed in the annular
flume using Lake Francis sediment in water with varying salinity were
analyzed in order to determine the combined effect of salinity and bed
shear stress on the settling rates of this sediment. The analysis
performed is described next.
The settling velocity at the time when 50% of the depositable
sediment had deposited, i.e. tsg. was determined for each experiment in
Range I in order to quantitatively evaluate the effect of salt
concentration, and possibly the bed shear stress, on the rate of
deposition. This particular value of the settling velocity, designated
as WggQ, was chosen for this analysis because it can be shown to be more
representative of the deposition rates in the time interval of interest
in numerical simulation (see Section 3.6.4. for discussion of this
aspect) than either the mean or the median value. The analysis was
performed in the following manner for the experiments in Range I (i.e.
1^ = 0.0 and 0.15 N/m''). Equation 3.28 was integrated and rearranged to
yield the given expression for W^jq:
C C
d ^0 ^50
W = An ( ) (3.49)
Pd^50 ^0%q
where C^q = suspension concentration at time t^Q. Values of W^^q
computed from Eq. 3.49 are shown in Fig. 3.49 superposed on Fig. 3.44,
with lines of equal shear stress drawn as indicated. The average value
of the initial concentration, Cq. in these experiments was 4.2 g/1 . No
consistent trend between and W rg (e.g. increasing values with
146
1.0
0.8
_ 0.6
1 i 111'
T 1 — r~i — ill!
E
E
>
o
o
_)
UJ
>
UJ
0.4
0.2'
0.
0.08
Q06
004
0.02
(N/m^)
• 0.0
* 0.015
Average Values
Owen s Data
Suspended Concentration
o
□
0.25
1 .0
4.0
16.0
32.0
g/f
g/^
J L
I I I
J I I I M
.0 20 4,0 6.0 8.0 10. 20.
SALINITY. S (PPT)
40. 60 80.100.
Fig. 3.49. Effect of Salinity and Bed Shear Stress on Settling
Velocity of Lake Francis Sediment.
147
increasing ^^ values for all salt concentrations) is apparent in Fig.
3.49. This observation suggests that W^gQ may be considered to be
invariant with respect to i^i^ in Range I.
Due to the limited data obtained, as well as the noted invariance
of W55Q with respect to t:^ for Range I, the values of W^gg for the two
values were averaged for each salt concentration. These average
values, "Ws50» ^^''^ plotted against salt concentration in Fig.
3.49. Such an averaging procedure was performed in order to further
investigate the effect of salt concentration on W^gg.
A power curve relationship between Vl^gg and the salinity, S, of the
following form was desired:
WgggCS.C) = A W^5q(35,C) (3.50)
where ¥55g(35,C) = KC" (K and n are defined in Section 3.6.1.) and A and
m are empirical constants. It was proposed that when S was less than
0.1 ppt, its value would be set equal to 0.1, so that W55g(S<0.1,C)
would be greater than zero and in fact W55g(S<0.1,C) = Wg5g(S=0.1,C) .
The value of V^gg at S = 35 ppt was utilized as the W55g(35,C) value.
Least squares linear regression analysis was used to determine if the
averaged settling velocities followed such a power curve. This analysis
gave the following values for A and m and the coefficient of
determination, r^: A = 0.57, m = 0.13 and r^ = 0.96. As indicated by
this r^' value, a good agreement was obtained between Eq. 3.50 and the
data. This confirms that, at least for these experiments and the
analysis method employed, the effect of salt concentration on W^gg in
Range I can be expressed by a power relationship of the form given in
Eq. 3.50.
148
The function given in Eq. 3.50 is incorporated into the deposition
rate expression for Range I and Range IIB as follows: Equation 3.50 is
used to evaluate the settling velocity as a function of the
concentration of dissolved salt and suspended sediment. Based on the
variety of relationships found between W^^g and C for several of these
experiments, the following general expressions for W^gQ in Range I and
for C < in Range II have been incorporated into the deposition
algorithm:
m
W m = A,.W .S for C < (3.51a)
SoU 1
I n, m^
^s50 " '^l*^ ^1 ^ *^ ^ ^2 (3.51b)
I n^ m.
^s50 " ^2 < C < (3.51c)
2 ^
AgD ( — 1) p
250 3"^ P C n m
W = ( ■) ( 1) S for C > C (3.51d)
pl.8 18V C3
where K^ = A^K^ for i=l,2, and W^,^ is the constant settling velocity for
C < Cj^. For concentrations greater than C^^ and less than C3, where
hindered settling begins, two different expressions of the form W^i^q =
K'C"s'" can be used to express the variation of W^^q with C and S. Two
relationships were revealed in this concentration range in several of
149
the deposition tests (e.g. see Fig. 3.50). These relationships are
permissible in the deposition/settling algorithm. If only one
relationship is revealed from the loglog plot of W^^q against C, C3
must be set equal to The values of ^^ and m^ apply for C < C2 while
A2 and m^ are for C2 ^ C < C3, and A3 and m3 apply for C > C3; if C3 =
C2, then A2 = ^l and m2 = m^. The values of K^^ and n^ apply for < C
< C2, while K2 and n2 are for < C < C3; if C3 = C2, then K^,^ = K2 and
n^ = n2. The values of K3 and n3 apply for C > C3.
For the depositionsalinity experiments in Range II (i.e. =
0.05, 0.1, 0.2, and 0.3 N/m ) the following analysis was performed.
*
Figure 3.51 shows the relationship between C^q and the salt
concentration for the values \ > \ . where Cg > 0. Based on the
mi n
nature of the equal curves in this figure and taking into
consideration the limited number of deposition tests performed at >
*
Tu , C is assumed to be invariant with respect to salinity.
''min
In Range IIB, the effect of salt concentration on the deposition
rate was evaluated in the following manner. The value of dC/dt at t=t5g
was determined for each experiment. Substituting t=t5Q into eq. 3.37
gives the following expression for the rate of deposition at tgg:
dC 0^3^ Co(l<q^
'50= (3.52)
/2^ a t
2 ^50
The value of t5g for each experiment was determined from the lognormal
*
plot of C versus t, and the value of 02 was evaluated according to
(Aitchison and Brown, 1957):
150
£
5
Q3
0.2
_ 0.10
0.08
>: 0.06
5 0.05
3 004
> 0.03
o
□ 0.02
H
H
LU
CO
0.01
0
^0.0 N/m
S O.ppt
T — r
2
"1 — r
1 III
02 0.4 0.6 0.8 1.0 2.0
CONCENTRATION, Cig/i)
4.0
Fig. 3.50. Settling Velocity Versus Suspension Concentration for
Deposition Test with Lake Francis Sediment.
0.6
=0.3 N/m
0.2 N/rrf
0.1 N/m^
0.05 N/m'
10 15 20 25 30
SALINITY. S { PPT)
35
Fig. 3.51. Variation of C^^ with Salinity and x^.
151
(3.53)
*
where tj^g and tg^ are the times at which C = 16% and 84%, respectively,
and likewise were determined from the lognormal plot. Due to the
limited data and observed invariance of (dC/dt)5Q with respect to t,^,
the values of (dC/dt)50 were averaged for each salt concentration. The
following functional relationship between T (given by Eq. 3.36) and the
salt concentration, S, was obtained using a linear regression analysis:
where b = 10.78. f = 0.33 and = 0.93. When S < 0.1 ppt, S is set
equal to 0.1 ppt.
The effect of salinity on the deposition rate expression for Range
IIB was incorporated by substituting Eq. 3.54 into Eq. 3.37. The
resulting value of the timerate of change of concentration, dC/dt, will
decrease monotonically at any given time with decreasing salinity, while
at the same time, the expression for C given by Eq. 3.35 still
approaches 0 as t^. This methodology of incorporating the effect of
T = Tog^QCf )BS^]
^50
(3.54)
*
salinity requires that the values of tgg and cr^, used in Eq. 3.37, be
evaluated at a salinity of 35 ppt.
152
3.6.4. Deposition Algorithm
Deposition of suspended cohesive sediment occurs when 1) the flow
is decelerating, i.e. Xj^(t+At) < ^^^(t), and 2) when tj^(t+At) < '^b^^^
VJhen these two conditions are satisfied at any node, the rate of
deposition is calculated as follows. The value of t^jj^^ is evaluated
using Eq. 3.48. Inasmuch as the lognormal relationship was found not
to be suitable for \ < 0.25, the minimum x^^^ value is set equal to
0.25. Therefore, Eqs. 3.28, 3.29 and 3.51a are always used to calculate
* *
the value of dC/dt for 0.0 ^ \ \ c (^^"9^ ^h^^Q Eqs. 3.28, 3.47 and
3.51a are used for Range IIA. The maximum allowable value for i^j^^^, is
1.00; therefore, Eq. 3.45 is used to determine dC/dt in Range IIB.
The amount of dry mass of sediment deposited per timestep element,
Mq, is determined according to
dC
Mr, = ~ At d^ (3.55)
° dt ^
where d_ = V2 (d (t)+d (t+At) ) and dC/dt is given by
1 dC dC
[— +— ] (3.56)
2 dt^ dt^^^^
in Range I and Range IIA, and
153
C * At 0.434 f3
— (1  C) [erf(Jln( ) + 1]
in Range IIB. In Eq. 3.57. e = \[Q{t) + e(t+At)), where 9 = C, tgQ,
Cgq and o^. Equation 3.57 was obtained by integration of Eq. 3.45
from t=0 to t=At. The thickness of the bed formed by Mq is calculated
in the bed formation algorithm using the procedure described in Section
3.3.2. The sink term in the governing equation (Eq. 3.5) is given by
Eq. 3.56 in Range I and Range IIA, and by Eq. 3.57 for Range IIB.
For unsteady flows, as occur in estuaries, the value of C^^, which
is the steady state value of the suspended sediment concentration found
in laboratory tests under steady flows, is assumed to be zero.
Nevertheless, the laboratory determined lognormal relationship for
dC/dt, as given by Eq. 3.57, is used for Range IIB in the deposition
algorithm for the following reasons. The timestep. At, used in the
estuarial sediment transport problems is typically of the order
0.1t5o<At<10t5g. Therefore, dC/dt (given by Eq. 3.57) is significantly
greater than zero at time At. This implies that after any time interval
At, the suspended sediment concentration, C(t+At), does not approach
Cp_, but is assumed to be equal to the following:
dC
C(t+At) = C(t) At (3.58)
dt
where dC/dt is given by Eq. 3.57. Thus, for unsteady flow conditions,
the rate of deposition is considered to be a function of Cgq (given by
Eqs. 3.323.34). Such a consideration is required for a realistic
154
interpretation of laboratory deposition test results for the purpose of
ascertaining the depositional rates in the unsteady estuarial
environment.
3.7. Consolidation
3.7.1. Description
In this section consolidation of a freshly deposited cohesive
sediment bed is described, followed by a review of some of the research
that has been conducted on the consolidation of sedimenting clays, and a
discussion on the possible correlation between the density and shear
strength of such soils.
As described previously, an estuarial sediment bed is formed when
deposited sediment particles and/or aggregates comprising a stationary
suspension begin to interact and form a soil which transmits an
effective stress by virtue of particle to particle contacts. The self
weight of the particles, as well as the deposition of additional
material brings the particles closer together by the expulsion of pore
water between the particles and/or aggregates. Thus, consolidation is
caused by the selfweight of the sediment particles (Parker and Lee,
1979). According to Parker and Lee, a soil is formed when the water
content of the sedimentwater suspension decreases to the fluid limit.
Unfortunately, there is not a unique water content value for cohesive
soils at which the suspension changes into soil (Been and Sills,
1981). This critical water content (i.e. fluid limit) is a function of
the initial water content of the suspension. Prior to soil formation,
i.e. for water contents above the fluid limit, the suspension is in a
stressfree state.
155
During the transition from suspension to soil, an extremely
compressible soil framework or skeleton develops (Been and Sills,
1981). The strains involved in this first stage of consolidation are
relatively large. For example, Parker and Lee (1979) assumed that
strain is onedimensional and found that "the strain involved in the
consolidation of an element of suspension, from say 10 g/1 (assume
particle density 2650 kg/m^, sea water density 1030 kg/m"^) to the
commonly observed density of 1200 kg/ni^ would be derived from an initial
solids fraction of 0.0038 and a final solids fraction of 0.1064 and show
a 96% strain in the element." Lee aiid Sills (1979) state that this
consolidation strain may continue for several days, or even months. The
straining and upward expulsion of pore water gradually decreases as the
soil skeleton continues to develop. Eventually this skeleton reaches a
state of equilibrium with the normal stress of the overlying sediment
(Parker and Lee, 1979).
During the early stages of consolidation the selfweight of the
soil mass near the bed surface is balanced by the seepage force induced
by the upward flow of pore water from the underlying sediment. As a
result, the effective stresses acting in the near surface region are
very small and in general are not measurable (Been and Sills, 1981). As
the soil continues to undergo selfweight consolidation and the upward
flux of pore water lessens, the selfweight of this near surface soil
gradually turns into an effective stress. This surface stress, and
indeed the stress throughout the soil may first crush the aggregate
structure of the soil and then crush the floes themselves.
Primary consolidation is defined to end when the excessive pore
water pressure has completely dissipated (Spangler and Handy, 1982).
156
Secondary consolidation, which may continue for many weeks or months, is
the result of plastic deformation of the soil under a constant
overburden.
Figure 3.52 shows the variation of the mean dry bed density, p,
with consolidation time for Avonmouth mud (Owen, 1977), commercial grade
kaolinite in salt water (S=35 ppt) (Parchure, 1980) and for kaolinite in
tap water (S=0 ppt) (Dixit, 1982). Noteworthy is the very rapid
increase in p in approximately the first 48 hours, after which the
increase was much less rapid, and the almost asymtopic approach to the
final mean bed density, p^. Figure 3.53 shows the variation of the
normalized mean bed density with consolidation time for the same three
mud beds. This figure shows that, at least for these three mud beds,
the timevariation of the degree of bed consolidation with time was
approximately equal.
Figure 3.54 shows the dimensionless relationship found between the
bed bulk density, Pg, and the depth below the bed surface, z^, with
consolidation times of the order of 48 hours for four natural muds.
Figures 3.55a and 3.55b show the dimensionless relationship found by
Dixit (1982) for T^^^ up to 24 hours and greater than 48 hours,
respectively, for kaolinite beds in tap water. Figures 3.563.58 show
the density profiles in Figs. 3.23.4 replotted on loglog paper. A
brief review of the studies on the consolidation of clays is given
below.
Terzaghi (1924) developed the first theory governing one
dimensional primary consolidation in soils. This theory was based on
several assumptions: 1) homogeneous soil, 2) onedimensional
compression, 3) onedimensional (vertical) flow, 4) the selfweight of
157
TIME (hrs)
Fig. 3.52, Variation of Mean Bed Density with Consolidation Time
(after Dixit, 1982).
■ 0 40 80 120 160
TIME (hrs)
Fig. 3.53. Variation of p/p with Consolidation Time (after
Dixit, 1982).
158
3.54. z^/H Versus p/p for Avonmouth, Brisbane, Grangemouth
and Belawan Muds (after Dixit, 1982).
159
Fig. 3.55. z^/H Versus p/p for Consolidation Times (a) Less Than 48
Hours and (b) Greater Than 48 Hours (after Dixit, 1982).
160
X
1
1.0
0.5
0.2
^ 4.17
o 8.33
o\ \
1 1
^ \ 1
0.3 0.5
1.0
p/p
2.0 3.0
Fig. 3.56. Normalized Bed Density Profiles for Thames Mud for Two
Different Consolidation Times.
1
X
1.0
0.5
0.2
Ql
005
1 1
<\
\Q
^\
Tjj(=2.78 days
S(ppt)
. 0
0 2
X 4
<^ 8
' 16
0 32
i
1 1
02
0.5
1.0
p/p
2.0
5.0
Fig. 3.57. Normalized Bed Density Profiles for Avonmouth Mud as a
Function of Salinity.
161
1.0
0.50
020
0.1
s ^
0.05
278 Days
17.3 ppt
16.7 g/^
■0 ■■ P ^
(m) (m) (kq/m^)
. 2.0 0.13 262
o 4.5 0.32 251
^ 70 0.58 211
a 10.0 075 194
0.02
0.0!
0.2
0.5
2.0 3.0 50
P / P
Fig. 3.58. Normalized Bed Density Profiles for Different Bed
Thicknesses .
162
the soil particles is neglected and 5) there is a constant relationship
between void ratio and permeability and between void ratio and effective
stress. The fifth assumption implies that the strain is assumed to be
infinitesimal with respect to the thickness of the soil layer. All
these assumptions limit the applicability of this theory to relatively
stiff thin layers at large depths. Therefore, it can not be used in
calculating the consolidation of soft sediment deposits, such as occur
in estuaries, in which large strains are not uncommon and the
relationship between void ratio and effective stress is not a constant
(Einsele et_al_., 1974).
Gibson (1967) developed the first completely general theory
of onedimensional primary consolidation of soils. The assumptions
inherent in this theory are: 1) the soil is saturated and consists of
an incompressible pore fluid and a compressible soil framework, 2) the
pore fluid velocities are governed by Darcy's law, and 3) there is a
unique relationship between the void ratio and soil permeability and
between the void ratio and effective stress. The governing equation for
this finite strain theory is
t
d k(e) ae 1 5 k{e) da de 1 Qe
_ [ .] + [ ] + = 0 (3.59)
de 1+e az s1 az (1+e) de az S1 at
w
where e = void ratio, Z = time independent material coordinate, which is
a measure of the volume of solid particles only, k = soil permeability,
S = soil specific weight, ^ = unit weight of the pore fluid and a' _
effective stress. This theory can be applied to thick soft sediment
163
deposits because it includes the effects of the selfweight of the
sediment particles, is independent of the degree of strain and includes
the constitutive relationships between e and k and between e and o
(Cargill, 1982).
Because of the nonlinearity of Eq. 3.59, general closed form
solutions have not yet been obtained. Therefore, numerical methods must
be used to solve this equation for e(Z,t) for given constitutive k and
a relationships and boundary conditions. Finite strain consolidation
computer programs which solve Eq. 3.59 using explicit finite difference
techniques have been developed by Gibson et__a2_. (1981), Schiffman and
Cargill (1981) and most recently by Cargill (1982). The required
constitutive relationships may be determined using any one of the
following devices: geotechnical centrifuge, stresscontrolled slurry
consolidometer, pore pressure probe and nuclear densitometer (Croce,
1982; Znidarcic, 1982). One important limitation of finite strain
consolidation theory is that since the effective stress at the surface
is assumed to be zero for all times, the void ratio there remains a
constant. This differs, as mentioned previously, from the observed
decrease in void ratio (which is directly related to p) at the
surface of soft sediment beds with time (Lee and Sills, 1981).
Lee and Sills (1981) assumed that the two constitutive
relationships were linear, and that the consolidation coefficient
dcJ
P^(l+p) de
(3.60)
164
was a constant and solved Eq. 3.59 analytically. They accounted for the
decrease 1n the void ratio at the soil surface with time by adding an
imaginary overburden of specified thickness on top of the actual bed.
Comparison of measured density profiles for a clayey silt with those
predicted by the analytical model yielded satisfactory results (Been and
Sills, 1981). However, inadequacies of the assumptions made by Lee and
Sills (1981) are reflected in these comparisons. Next, a discussion is
presented on the shear strength characteristics of cohesive soils and
the possible correlation between the density and shear strength of these
soils.
The shear strength of clays are due to the frictional resistance
and interlocking between particles (physical component), and
interparticle forces (physicochemical component) (Karcz and Shanmugam,
1974; Parchure, 1980). Taylor (1948) states that some of the factors
which affect the shear strength of clays are: type of clay mineral,
water content, consolidation tine, stress history, degree of sample
disturbance, chemical bonding (cohesion), anisotropy and exchange of
cations. Consolidation results in increasing bed density and shear
strength (Hanzawa and Kishida, 1981). Figure 3.59 shows the increase in
the shear strength profile with consolidation time for flowdeposited
kaolinite beds in tap v/ater.
As mentioned previously, the nature and effect of consolidation on
the shear strength profiles of cohesive sediment beds are not well known
at present, and what limited information has been obtained is often
contradictory (Parchure, 1980). Several researchers have attempted to
establish a correlation between the bed density and shear strength of
clay beds in order to implicitly evaluate the effect of consolidation on
i^c(Z). The results of these efforts are summarized below.
165
Fig. 3.59. Variation of (Z^) with for Various Consolidation
Periods (after Dixit, 1982),
I
i
166
Figure 3.60 shows the correlation Owen (1970) found between the dry
sediment density and shear strength for statically deposited beds of
Avonmouth mud. Least squares analysis of the data plotted in this
figure gave a slope of 2.44 and a coefficient of determination of r^ =
0.83. The power expression relating and p is of the form
\ = ^ (3.61)
with a = 5.85x10"^ and P = 2.44. Owen considered that the correlation
obtained between and p was satisfactory, considering the experimental
error involved in the measurement of both these parameters.
Thorn and Parsons (1980) likewise found a power relationship
between and the dry sediment density at the bed surface, „ , for
Grangemouth, Belawan and Brisbane muds in saline water. They obtained
values of 5.42 x 10 and 2.28 for a and p, respectively. In an earlier
study, however, these researchers observed no significant correlation
between and p^ for Grangemouth mud (Thorn and Parsons, 1977), while
Thorn (1981) obtained a linear relationship between i^ and p^ for mud
from Scheldt, Belgium in 5 ppt water.
Bain (1981) found a relationship of the following form between e
and for Mersey and Grangemouth muds:
^ P
(3.62)
167
10.0
X
UJ
m
cr
<
LU
X
m
5.0
1.0
05
1 1 1
c
1
s
1 !
d
1 1 /I ■
(mg/ )
(m)
~ o 16.290
32.9
10.06
 • 15.520
17.0
9.38
▼ 17.475
17.8
6.98
J3 /
^ /
■ 1 1 .do\J
lb. /
/I HA
4.o4
7
9 6.705
2.7
9.73
7.288
4.6
9.74
/ —
O 8.392
8.8
a76 ^
a 10.272
16.8
1002 ^
X /
6.866
33.3
9.72
A 6 810
0 8
9 74'
h
▼
' •Onr/
■ • T
^ /

• / X

▼ ▼ p
«
a /
/ oo
/ °
h
a /
o
ft
X
1 1 /
1
1 1
1 1 1
60 80 100 150 250
DENSITY ^(kg/m^)
350
Fig. 3.60. Correlation of Bed Shear Strength with Bed Density
(after Owen, 1970).
168
with C = 2 X 10^'^ N/m^ and ^ = 18.3 for f^ersey mud, and C = 8 x 10'' and
^ = 6.1 for Grangemouth mud. The diversity of these relationships
serves to emphasize that such properties of cohesive sediment soils must
be established for each sediment studied. A description of the
consolidation algorithm is given in the next section.
3.7.2. Consolidation Algorithm
Consolidation of the deposited sediment bed is accounted for by
increasing the bed density and shear strength with time. These
calculations are made on an elementbyelement basis. The description
given for the consolidation algorithm developed as part of this study is
for any one given element.
Consolidation is considered to begin after the bed formation
process is complete, at which time the bed thickness will be maximum.
As described in Section 3.3.2 and Appendix D, Section D.2, the
discretized density profile at a consolidation time, T^^., of a certain
magnitude T^^^^ (selected to be equal to two hours as measured in
laboratory tests) is used in the bed formation routine to form the bed
structure resulting from deposition of a given mass of sediment. The
consolidation period for a bed begins the first time step during which
no deposition is predicted to occur.
After Tj^j hours of consolidation (i.e. 2 hrs.) the stationary
suspension present on top of the bed becomes part of the partially
consolidated bed, and therefore would undergo resuspension if subjected
to an excess shear stress. The dry sediment mass of the stationary
suspension is first determined, and then the bed formation routine is
used to evaluate the thickness and structure of the partially
169
consolidated bed formed by this mass. This procedure accounts for the
observation that after approximately two hours of consolidation a
cohesive sediment bed undergoes resuspension when subjected to an excess
shear, whereas for T^^^. < 2 hours the bed mass erodes when subjected to
the same shear (Dixit, 1982). Should further deposition occur for T^^^ <
2 hours, the value of T^^. is reset to zero, and the new bed formed is
evaluated using the crushing procedure in the bed formation routine.
The increase in bed density and shear strength with time is
simulated to begin at T^^^, = 2 hours. This is the reason why the
measured bed density profile at T^^ = 2 hours is used in the bed
formation algorithm. The procedure for evaluating p{z^,t) is given
below.
1) First, the final mean bed density, is determined. ~„ and the
time, Tj^^, at which p = Pa, have been shown to vary linearly with C^,
which is defined to be the suspension concentration that would result if
the entire partially consolidated bed was resuspended (Owen, 1970).
is given by the following:
pH
(3.63)
where p and H = mean bed density and bed thickness, respectively, of the
partially consolidated bed at the end of the previous time step (i.e.
t=tAt, where At = time step). The equations used for ~^ and T^ are
170
P = (p ) + aC
"0 0
(3.64)
T . = (T , ) + bC
dc^ dc^'^ 0
(3.65)
where (Poo)q and (T^^^ )q are the extrapolated (hypothetical) intercept
values for = 0. These two parameters and a and b are empirical
coefficients that must be determined experimentally. The value Pa, is
defined to the value at t = T^^ at which the following criterion is
satisfied:
p(t)  P(tAt)
< 10
3
P(t)
(3.66)
with At taken to be 24 hours.
2) The value of p(t = T^^) is determined next. The following
relationship for p is indicated by Fig. 3.53:
P(Tdc^/P»= 1  f*exp(pT^,/T^^ ) (3.67)
CO
Least squares analysis of the three sets of data plotted in Fig. 3.53
gave f = 0.845, p = 6.576 and for the coefficient of determination r^ =
0.993.
3) The bed density profile P(z5,T^j.) is next determined. The density
profiles given in Figs. 3.543.58 can be expressed as
171
Hz.
= A{ )'
(3.68)
where the values of A and B are functions of time for T^^ < 48 hours,
and constants for T^^ > 48 hours. This variation of the bed density
profile with time is depicted in Fig. 3.61. The value of z' = (HZ[j)/H
below which A and B are invariant with respect to time is defined to be
^max* ^^''^ r^d^r\% that for T^^^ > 48 hours and for T^^, < 48 hours with z'
^ ^max' P''^b^ ^H 3.68 with A and B equal to the respective
values for T^^, > 48 hours, while for T^^ < 48 hours and z'^^^ < z' < 1.0,
pfZfj) is given by Eq. 3.68 with the values of A and B being functions of
time. The values of A, B and z^jj^^ determined for the density profiles
in Fig. 3.55a are given in Table 3.3.
Table 3.3
Variation of Empirical Coefficients
the Relationship Between p{z^) and T^^^.
Tj^( hours)
7'
max
2 0.36 1.40 0.43
5 0.48 0.72 0.60
11 0.62 0.45 0.76
24 0.66 0.50 0.84
>48 0.80 0.29 1.0
Fig. 3.61. Variation of p(z, ) with T,^ Incorporated in Consolidation
Algorithm. °
173
Least square analysis of the data given in the above table revealed that
all three parameters. A, B and z^j^^^^, varied with J^^ according to
® " '^^dc'' "^dc ^ ^^^^^ ^^'^^^
with D = 0.32, 1.71 and 0.39, G = 0.24, 0.49 and 0.24, and r^ = 0.96,
0.95 and 0.96 for 9 = A, B and z,,ax» '"'^specti vely.
Figure 3.58 shows a bed density profile for which p was measured
below z' = 0.05. Based on the best fit line drawn through the data
points in this figure, it is assumed for 0.0 < z' < 0.05, p(z^) is equal
to the constant value p(Z(^=0.95H). This extrapolation of the density
profile down to the bottom of the bed, = H, is required for two
reasons: 1) in order to determine 9{z^=H), since the power law given
by Eq. 3.68 cannot be evaluated at = H, and 2) in order to insure
conservation of sediment mass in the bed. To summarize, p{z\^) is
evaluated at a particular J^^ value as follows. For T^^ < 48 hours:
'^"^^ n Hz, 1.71T,
= 0.32T^^02^^) dc (3^^^^
H
0 24 "^b 1.71(48)1^^
= 0.32(48)"^^ ) (3.71)
H
174
for 0.05 < z < z^^',
max
^^^b^ 0.24 1.71(48)"°*^^
= 0.32(48) (0.05) (3.72)
= 1.75 for 0.0 < z < 0.05
0.24
where z^^^ = 0.39 T^^ . For T^^^. > 48 hours, Eqs. 3.71 and 3.72 are
used with z'^^^ = 1.0. The discretized bed density profile used in the
layered bed model is changed using Eqs. 3.703.72 to reflect the
increase in bed density due to consolidation.
4) The thickness of the bed, H, is reduced to account for the expulsion
of pore water during consolidation, and to insure that the mass of
sediment in the bed is conserved. The rate of change of H with time is
given by
dH H dp
dt=dt (■^•73)
P
Using first order finite difference approximations for the two
derivatives, Eq. 3.73 becomes
p(tAt)  p(t)
H(t) = H(tAt) {l+2( )} (3.74)
p(t) + p(tAt)
175
where H(t) = bed thickness at the current time step and H(tAt) = bed
thickness at the previous time step. The thickness of each bed layer is
adjusted as follows:
H{t)H(tAt)
T.(t) = T.(tAt){H } (3.75)
^ ^ H(tAt)
where T^ = thickness of the il!l bed layer.
5) If further deposition occurs when T^^ > 2 hours, a new sediment bed
is formed on top of the existing partially consolidated bed. To
simulate the occurrence of such repeated periods of deposition, as
typically occurs in estuaries due to the oscillating tidal flow, the new
deposit portion of the bed model is further divided into a finite number
of strata (Fig. 3.62). The top stratum may be composed of a stationary
suspension and partially consolidated bed, whereas the buried strata are
composed of just partially consolidated beds. The degree of
consolidation of a particular stratum (in relation to that of the other
strata) is accounted for by using a separate T^^ for each stratum. The
bed density profile for the 1— stratum as a function of T^^.., where the
subsubscript i refers to the i~ stratum, is determined as follows.
Step 1 is performed with H = total bed thickness of all strata and 7 =
mean bed density of all strata. Step 2 is repeated for each stratum.
Thus a separate value of p is determined for each stratum. Steps 3 and
4 are likewise repeated for all the strata, with the total bed thickness
used for H and a separate value of p used for each stratum.
176
Strata
X
®
iz
X
dc
nr
ir
'dc
IE
I ©
Tdci
Bed Surface
Settled Bed
z = I.O
UNO
PCND ^
PCND
PCND
PCND
•z =0.0
"^dc "^dc ^ "^dc Tiip
ULj ULjj UCjjj UCj2
Fig. 3.52. Bed Schemati zation Used in Bed FormationConsolidation
Algorithms .
177
Due to the extremely limited number of studies on the nature of
shear strength profiles in cohesive sediment beds, the variation of the
bed shear strength profile "^^^^b^ ^^'^'^ "'"dc ^'^ determined indirectly by
use of a functional relationship between p and x^. The relationship
found between these two parameters (Eq. 3.61) by Owen (1970) and Thorn
and Parsons (1980) is used in the consolidation algorithm to account for
the increase in with increasing T^^^.
The empirical coefficients used in the consolidation algorithm
(i.e. a, p, (p„)g, (T(J(;^)q, a, b, f, p. A, B, and z'^^^) must be
determined by performing laboratory consolidation tests. A
brief description of a test procedure is given in Appendix D, Section
D.2.
CHAPTER IV
MODEL DEVELOPMENT
4.1. Introductory Note
This chapter begins with a review and evaluation of previous
cohesive sediment transport models, then follows with descriptions of
the cohesive sediment transport model developed during this study, the
finite element formulation used in the model, and lastly, the
convergence and stability characteristics of the model.
4.2. Review of Previous Models
One of the first cohesive sediment transport models was developed
by Odd and Owen (1972). This was a twolayered, onedimensional coupled
model which simulated both the tidal flow and mud transport in a well
mixed estuary. The two layers were of unequal thickness, with uniform
properties (e.g. flow velocity, suspension concentration) assumed for
each layer. A rectangular flow crosssection was also assumed. The
equations of motion and continuity for each layer were solved using a
finitedifference formulation, while the advectiondiffusion equations
governing the transport of suspended sediments in two layers were solved
using the method of characteristics. Erosion and deposition of sediment
were simulated in this model.
O'Connor and Zein (1974) developed an uncoupled twodimensional,
laterally averaged suspended sediment model which solves the advection
178
179
diffusion equation using an implicit finitedifference method.
Horizontal eddy diffusion is neglected, as is the assumed negligible
vertical water motion in comparison with the sediment settling
velocity. The settling and erosion of sediment was accounted for in
some of the described model applications by modifying the vertical
sediment diffusion coefficient. The model is strictly applicable to
quasisteady depth and flow conditions. It was later modified to
include unsteady (tidal) flow conditions by O'Connor (1975).
Ariathurai (1974) and Ariathurai and Krone (1976) developed an un
coupled twodimensional, depthaveraged sediment transport model which
uses the finite element method to solve the advectiondispersion
equation. This model simulates the erosion, transport, aggregation and
deposition of suspended cohesive sediments. Aggregation is accounted
for by determining the sediment settling velocity as a function of the
suspension concentration. Required data include the twodimensional,
depthaveraged velocity field, dispersion coefficients, and the sediment
settling and erosion properties. Ariathurai et al_. (1977) modififed
this model to solve the twodimensional, laterally averaged suspended
sediment transport problem. This latter model was verified using field
observations in the Savannah River Estuary.
Kuo^al_. (1978) developed a twodimensional, laterallyaveraged,
coupled model which simulates the motion of water and suspended sediment
in the turbidity maximum of an estuary. The vertical dimension is
divided into a number of layers, and a finite difference method is used
to solve the equations of motion, continuity and sediment mass balance
for each layer. Erosion and deposition are accounted for in the mass
balance equation for the bottom layer.
180
Koutitas and O'Connor (1980) developed a threedimensional
suspended sediment transport model which solves the advectiondiffusion
equation using a mixed finite differencefinite element method. Central
finite differences are used in the horizontal directions while linear
finite elements are used in the vertical direction. The two horizontal
turbulent diffusion coefficients and the vertical water velocity are
assumed negligible in comparison with vertical eddy diffusivity and
sediment settling velocity, respectively. The source/sink term which
accounts for the erosion and deposition of sediment is not included in
the governing equation.
Cole and Miles (1983) describe a twodimensional, depthaveraged
model of mud transport which solves the advectiondispersion transport
equation by a finite difference method. Deposition and dispersive
transport are simulated, but erosion is not.
None of these fine sediment transport models consider the following
two factors. 1) Consolidation of the mud bed and the effect this has on
the erodibility when the bed is subjected to an excess bed shear
stress. 2) The effect of salinity variation (e.g. in the mixing zone
between fresh and sea water in estuaries) on the processes of erosion
and deposition of cohesive sediments in a turbulent flow field, since
the empirical laws used to determine the rates of erosion and deposition
were derived using empirical evidence from laboratory experiments
conducted in natural or artificial sea water. In addition, the
empirical laws of erosion and deposition used in these models cannot be
considered to be "the stateoftheart" even for sea water, as a
considerable number of laboratory tests conducted since these laws were
proposed have revealed new evidence on the erosional and depositional
181
behavior of cohesive sediments. For example, the empirical erosion rate
expression used in the existing models is given by Eq. 3.13, which, as
described in Chapter III, has been found to be applicable only to
settled beds and not to partially consolidated beds. Likewise, the
empirical deposition rate expression used in these models (Eq. 3.50) is
limited to only a small percentage (e.g. approximately 20% for kaolinite
in tap water) of the bed shear stress range over which subsequent
laboratory tests have shown that cohesive sediments deposit under steady
flow conditions (Mehta, 1973).
4.3. Model Description
The cohesive sediment transport model, referred to hereafter as
CSTHH, developed during this study is a time varying, twodimensional,
uncoupled finite element model that is capable of predicting the
horizontal and temporal variations in the depthaveraged suspended
concentrations of cohesive sediments and bed surface elevations in an
estuary, coastal waterway or river. In addition, it can be used to
predict the steadystate or unsteady transport of any conservative
substance or nonconservative constituent, if the reaction rates are
known. CSTMH simulates the advective and dispersive transport of
suspended or dissolved constituents, the aggregation, deposition and
erosion of cohesive sediments to and from the bed, respectively, and the
consolidation of the bed.
CSTMH is composed of the algorithms and layered bed model
developed in Chapter III integrated into a modi_fied version of the
finite element solution routine developed by Ariathurai (1974). A
description of this finite element formulation and the modifications
182
made to it is given in the next section. A synopsis of the operations
performed by CSTMH during each time step is given below. The flow
chart given in Appendix C, Section C.3 depicts the stepbystep solution
procedure incorporated in CSTMH.
The average bed shear stress induced by the turbulent flow velocity
of the suspending fluid is calculated for each element. Then the amount
of sediment, if any, that was deposited onto or resuspended from the bed
in each element during the previous time step is determined using the
deposition and erosion algorithms, respectively. The dispersion
algorithm then calculates the values of the four components of the two
dimensional sediment dispersivity tensor. Using these values and the
prescribed velocity field and concentration boundary conditions, Eq. 3.5
is solved for the suspended sediment concentration at each node for the
next time step. The new bed elevation in each element is determined by
adding or subtracting the thickness of sediment deposited onto or
resuspended from, respectively, the bed profile that existed during the
previous time step. Lastly, the consolidation algorithm calculates for
each element the increase in bed density and shear strength and the
decrease in bed thickness due to consolidation during the previous time
step.
The following five types of data are required to apply CSTMH to a
particular water body: 1) input/output and transient control
parameters, 2) finite element grid of the system to be modeled, 3) two
dimensional depthaveraged velocity and salinity fields, 4)
concentration initial and boundary conditions and 5) properties of the
cohesive sediments in the water body to be modeled which characterize
their erosion, deposition and consolidation. The user's manual in
183
Appendix C, Section C.4 lists the required information in each of these
five data groups. A brief description of the five groups is given
below.
The required input/output and transient control parameters include
the time step size, the degree of implicitness used in solving the
temporal problem, and code arrays which specify at which time steps new
values of various parameters (e.g. depth of flow) are read in. These
transient code arrays also specify the type of output (e.g. nodal
concentrations and/or discretized bed profiles), if any, required at
each time step.
The finite element grid of the water body to be modeled is defined
by the number of elements and nodes the water body is divided into, the
two horizontal coordinates of each node point, and the number of nodes
which form each element. Quadratic and/or triangular elements with
curved sides may be used in CSTMH. A finite element grid generating
program is a helpful tool in generating and modifying a finite element
grid. This is especially true when a relatively large body of water
(e.g. Tampa Bay) is being modeled. The advantages of using the finite
element method over the more conventional finite difference method in
estuaries and other similar bodies of water are delineated in the next
section.
The velocity field is defined by the two horizontal components of
the depthaveraged flow velocity and the depth of flow at each node and
time step. The only practical methods available today to determine the
velocity field in the detail required by an uncoupled transport model
such as CSTMH to model an estuary or other prototype water body are
physical and mathematical models. The advantages and disadvantages of
184
each are well documented, and therefore will not be discussed here.
Numerous twodimensional hydrodynamic mathematical models have been
developed; these are likewise well documented in engineering
literature. The FESWMS hydrodynamic model developed by the Water
Resources Division of the U.S. Geological Survey is particularly well
suited for modeling the velocity and salinity fields because this model
uses the same basic finite element formulation (i.e. isoparametric
quadrilateral and/or triangular elements with parabolic sides) that is
used in CSTMH. Therefore, the same grid can be used in both models.
The salinity (i.e. density) field needs to be determined for water
bodies where spatial and/or temporal variations in the salinity occur,
in order to model the effects that variations in the salinity have on
the erosion and deposition characteristics of cohesive sediments. A
description of the method used in CSTMH to evaluate the bed shear
stress over a cohesive sediment bed using the two depthaveraged
horizontal velocity components is given later in this section.
The depthaveraged suspended sediment concentration must be
specified at each node at the start of the modeling effort (initial
conditions). Boundary conditions (i.e. depthaveraged suspension
concentrations or normal concentration flux) are required for all
external v/ater boundaries of the system being modeled. For the nodes
which define such external water boundaries at which no concentration
boundary conditions are given, CSTMH assumes that the spatial
concentration gradient is zero. The boundary conditions at the free
water surface and the bottom are expressed by Eqs. A. 35 and A. 36,
respectively. Equation A. 35 expresses that there can be no net rate of
sediment transport across the free water surface. The bottom boundary
185
condition (Eq. A. 36) expresses that eroded sediment material is
transported in the vertical direction away from the bed by turbulent
diffusion and that deposited sediment becomes part of the bed.
Equations A. 35 and A. 36 are actually included in the governing equation
(Eq. 3.5) since it is vertically integrated from the bottom to the free
water surface (see Appendix A).
The sediment parameters which prescribe the erosion, deposition,
bed formation and consolidation characteristics of the sediment in the
water body to be modeled are described in Chapter III, Sections 3.3,
3.4, 3.6 and 3.7 and in Appendix C, Section C.4. The field data
collection and laboratory sediment testing programs recommended for
obtaining the data required by CSTMH are described in Appendix D.
Descriptions of some additional functions incorporated in CSTMH are
given below.
The settling velocity in Range IC is a function of the kinematic
viscosity, v (see Eq. 3.59d). The following equation for v as a
function of the mean water temperature, (which is read in the fifth data
set above) was determined with data obtained from Bolz and Tuve (1976),
using least squares linear regression analysis:
V = 1.7017x10"^ • exp(0.0251T ) (4.1)
w
where T^ is the mean water temperature in degrees Celsius and v has
units of m /s. The coefficient of determination for this equation is
2
r =0.994, which indicates a good agreement between Eq. 4.1 and the data.
The density of the suspending fluid (in Kg/rn^) is calculated at
each node as a function of T^ and nodal salinity value using the
following empirical equation (Wilson and Bradley, 1968):
186
P = 1000. 0*(0. 702 + 100. 0*(17. 5273 + O.llOlT 
w w
0.000639T^  0.039986S 0.000107T S)
w w
(5881.913 + 37.592T  0.34395T ^ +
w w
2.2524S)"^) (4.2)
where S = salinity in ppt. Incorporating, in addition, the effect of
the suspended sediment on the local water density gives (MacArthur,
1979)
e,w w s w
'^s
where Pg ^ ~ effective local water density = f(T^,S,C), and p^^ is given
by Eq. 4.2. At each time step where nodal salinity values change, new
nodal values of p^ and Pg ^ are calculated. At each time step where
nodal concentrations change, new nodal values of pg ^ are determined.
The values of pg ^ are used in CSTMH to calculate the nodal values of
the bed shear stress tj^, while the values of p^ are used in determining
the nodal values of the dry density, p, which are used in the erosion,
bed formation and consolidation algorithms.
The bed shear stress is calculated at every node using the
following relationship between and u^ = friction velocity:
\ = Pe,w"f^
(4.4)
187
Modal values for are determined as follows. The magnitude of the
depthaveraged velocity vector, U, at each nodal point is given by
 2 2^/2
U = (u^ + v^) (4.5)
where u and v are the two depthaveraged velocity components given as
input data. The vertical velocity profile for a fully developed two
dimensional, bounded shear flow over a cohesive sediment bed, which
Mehta (1973) and Gust (1976) have found to be hydrodynamically smooth,
is given by (Chri stensen, 1977)
U(2) •^f^
= 5.5 + 2.5 In ( 5.29)
(4.6)
where U(z) is the horizontal velocity component at an elevation z above
the bottom. Integrating Eq. 4.5 over the local depth of flow, d, gives
U
— = 2.5 In (
3.32 u^d
 17.56)
(4.7)
Using the value of U determined by Eq. 4.5, a NewtonRaphson iteration
scheme is used to iterate for the value of u^. Then tj^ is determined
using Eq. 4.4.
A description of the FORTRAN computer program of CSTMH is given in
Appendix C. In Sections C.l and C.2, the functions of the main program
188
and subroutines are respectively described. A flow chart and user's
manual are given in Sections C.3 and C.4, respectively. In the next
section, the finite element routine used is described.
4.4. Finite Element Formulation
4.4.1. Introductory Mote
The finite element method has been used to solve the governing
equation (Eq. 3.5). This method is a numerical analysis technique for
obtaining approximate solutions of differential equations. The
discretization procedures used reduce the equation to be solved to one
with a finite number of dependent variables by dividing the continuous
solution domain into a number of elements and by expressing the
dependent variable in terms of approximating interpolation (i.e. shape)
functions within each element. The values of the dependent variable at
node points are used to define the interpolation functions. Node points
are usually located on the boundaries of elements and are used to define
the connection between adjacent elements. The number and location of
the node points must be chosen such that continuity of the dependent
variable across common boundaries of adjacent elements is achieved
(Zienkiewicz, 1977). The behavior of the dependent variable within each
element is defined by the values of the dependent variable at the nodes
and the shape function. Then the error which results from the use of
the approximate dependent variable at each node in Eq. 3.5 is
minimized,, This procedure results in a set of simultaneous equations
which are solved for the unknown nodal dependent variables at the next
time step,, A detailed description of the method is presented by
Zienkiewicz (1977).
189
This method is preferred over the finite difference method because
derivative boundary conditions do not require special treatment in the
finite element method as they do in the former. It is a particularly
advantageous method to use in estuarial transport problems because of
the ability to use arbitrarily shaped elements.
Quadrilateral and/or triangular elements may be used in CSTMH in
which a quadratic function approximation is used to describe both the
intraelement spatial variation of the geometry and suspended sediment
concentration. Therefore, the elements are isoparametric and may have
curved sides.
4.4.2. Shape Functions
The global and local element coordinate systems are shown in Figs.
4.1a and 4.1b, respectively. The global xy coordinate system is
continuous over the entire solution domain, while the local element ?,ri
coordinate system applies only within an element. The local coordinate
systems for a quadratic quadrilateral and quadratic right triangular
element are shown in Fig. 4.1b. The local or area coordinates for the
corner nodes 1, 3 and 5 of the triangular element are (0,0), (1,0) and
(0,1), while those for the corner nodes 1, 3, 5 and 7 of the
quadrilateral element are (1,1), (1,1), (1,1) and (1,1).
Because three nodes are used along each edge of the triangular and
quadrilateral elements, quadratic shape functions are required. The
quadratic shape functions in CSTMH determine the values of both the
dependent variable, C, and the element geometry. Thus, the elements are
isoparametric (Zienkiewicz, 1977). There is one shape function, N^, for
every node in a given element. Thus, for triangular elements there are
190
7 6 5
I 2 3
(a) Global Coordinates
5(0.1)
(b) Local Coordinates
Fig. 4.1, Global and Local Coordinates
191
six shape functions while for quadrilateral elements there are eight.
The shape functions are functions of the local coordinates C and ti and
the values of C and ri at the nodal points. The quadratic shape
functions for quadrilateral and triangular elements are given in Table
4.1. The parameters C^ and ti^. in this table are the nodal
coordinates. For example, for a quadrilateral element, r\. = i, i
for node 1, while for a triangular element, 5^, t\. = 1, 0 for node 3.
The dependent variable, C, is approximated as the following
function of the unknown nodal point concentrations, C^ , and the shape
functions,, N:
i=n
C. = 2 N.C.
J i=l ' '
(4.8)
where Cj = approximate suspended sediment concentration at any location
inside the jth element, and n = number of nodes forming the jth element.
Likewise, the global coordinates x and y are approximated as the
following functions of the global nodal point coordinates, x^ and y^,
and the shape functions, N:
i=n
X = 2 N.X.
i = l ^ ^
(4.9)
192
Table 4.1
Quadratic Shape Functions
Quadrilateral Element
Shape Function Node Number
]. = (l+^C.)(l+TlTl.)(^.+TlTl.l)/4
Corner Nodes 1,3,5,7
M^. = (l^^)(l+TiTi^. )/2 Midsection Nodes 2,6
N^. = (1Ti^)(l+?C. )/2 Midsection Nodes 4,8
Triangular Element
Shape Function Node Number
N. = 4^.1,, (1Ti^. )(l€^.Ti^. ) Midsection Node 2
N^. = 4 l.r]./{l.+r].) Midsection Node 4
= 4 r]^{ll.r].){ll.) Midsection Node 6
N. = Ul^n.){l2{l.+r\.2l.r\.)} Corner Node 1
N. = 2lA/22r]^l.+r].^+^.l.+r]./{l.+r].)} Corner Node 3
2
^1 = ^T^l. Ti . +1. +C. Ti. +C. / ( 5. +n. ) } Corner Node 5
193
The shape functions are used for two additional purposes: 1) to
transform from the global coordinate system to the local element
coordinate system and 2) to transform the derivatives of C with respect
to X and y to the local element coordinates. To perform these
transformations, the derivatives of with respect to x and y are
needed. These are derived below.
The derivatives of N^ with respect to x and y are given in terms of
the derivatives of N^ with respect to the local coordinates C and t)
using the chain rule of partial differentiation (Zienkiewicz, 1977)
aN. ay aN. ay aw.
ax ari a^ a^ an
= [— • • — 1 ' iJf^ (4.10)
aN^. ax aN^. ax m.
— = C— • • — } ' iJi'^ (4.11)
dy aC dri at) a^
where Jl is the Jacobian given by
1J =
ax
ay
a^
ac
ax
A
ay
ari
(4.12)
The derivatives of N^. with respect to I and ri for both quadrilateral and
triangular elements are given in Table 4.2.
194
Table 4.2
Derivative of Shape Functions
Derivative
Quadrilateral Element
Node Number
^^.(l+TlTi^.)(2^S^.+r)ri^.)/4
Corner Nodes 1,3,5,7
Midsection Nodes 2,6
^i,.(lri^)/2
1
Midsection Nodes 4,8
n^.(l+CC.)(2TiTi.+^^.)/4
Corner Nodes 1,3,5,7
^. (lr)/2
Midsection Nodes 2,6
Derivative
Midsection Nodes 4,8
Triangular Element
(6 = Kronecker Delta)
Node Number
4[l2(C+Ti.)+2^.Ti+n.^]6.^.
4Tl.[lH./(C.+ri.)]/(^.+n.)6.^.
4Ti.(22C...)6..
( 3+45 . +8T1 . 81. ri. 471^ ) 6. .
{N„ / ^ 2C . [  1+Ti . +n / ( 5, +ri . ) 2 ] } 6
2ri. [2+21 .+T1 .+1/ ( 6 .+n . ) +C. / ( 5.+ti. ) ^1 6. .
45.(25.2..)6..
4?^.[l+n./(C.+n.)]/(5.+ri.)6.^.
4(l?.)(l5.2Ti.)6.^.
{3+4Ti.+8C.8C.ri.4C^?)6_
2C^ [2+2T1 . +C . +1 / ( I, +n . ) +n . / { I. +T1 . ) ^ ] 6 . ^.
{N5/T152T1. [1+5. +C. / ( 5. +T1. ) 2] I.
2
4
6
1
3
5
2
4
6
1
3
5
195
The four components of the Jacobian given by Eq. 4.12 are equal to
— = 2 X. — = Z X.
i=i ac ^ an i=i an ^
(4.13)
ay aM 5y . aN .
i=n 1 i=n 1
ac 1=1 ac an i=i an
Likewise, the derivatives of C with respect to C and n are equal to
ac . „ aw. ac . an.
i=n 1 i=n 1
— = 2 C. — = Z c, (4.14)
ac i=i ac ^ an i=i an ^
4.4.3. The Galerkin Weighted Residual Method
The Galerkin weighted residual method has been used to solve the
governing equation. This method requires that the summation of weighted
residuals over the entire solution domain be equal to zero when the
shape functions are used for the nodal weighting factors. The residual,
r, results from applying the governing equation to the element subdomain
using the approximate suspension concentration C instead of the actual
concentration C. In order for C to satisfy all the stipulated boundary
conditions, the sum of the normal concentration fluxes from adjacent
elements and any source or sink must be equal to zero on all internal
and external boundaries in the solution domain. This condition may be
expressed mathematically as (Ariathurai and Krone, 1976):
q^ + q^ + q^
196
= 0 i=l....,NL
(4.15)
where
q = outward normal flux from one element
= inward normal flux from adjacent element
qf = normal flux from source/sink on the ith boundary
NL = number of element interfaces and external boundaries
The formulation of the Galerkin method can be expressed mathematically
as
j=NE k=NL
2 / H.rdA + 2 / N.RdC = 0 (4.16)
j=l A k=l C ^
where Ag = element subdomain, R = residual which results from the use of
C in Eq. 4.15 and 5 = variable length along the k^ boundary.
The governing equation used by Ariathurai (1974) was Eq. 3.5 with
the offdiagonal dispersion coefficients, and D„„, equal to zero.
This formulation was modified to include the terms involving D^^y and
Dyj^. The following development is for the modified formulation
incorporated in CSTMH.
Substituting the expressions for r and R into Eq. 4.16 yields
j=NE ac ac ac a ac
2 //. N . { — + u — + v (D — +
j=i '^a at ax ay ax ^^ax
197
ac a ac ac
D — ) (D — + D — )  S } dxdy +
^ydy dy >^ydx yyay
k=NL
2 (q^ + q + q^) dC = 0
k=l
(4.17)
The term Q = _L + S in this equation is taken to be an instantaneous
at
constant. This approach, which transforms Eq. 3.5 into an elliptic
equation, results in a more efficient computational scheme.
Using Pick's law, the fluxes in Eq. 4.15 are given by
dc ac ac ac
(D — + D — ) n + (D — + D — ) n
XX 3^ xy^y X yx^^ yy^^ y
(4.18)
where n^, ny = x and y components of the outward normal to the
boundary surface of Ag.
The second derivative terms in Eq. 4.17 require continuity of the
first derivatives to insure convergence. This would involve solving for
the unknown first derivatives of C. To avoid this added complication,
the second derivatives are transformed to first derivatives using the
divergence theorem. The first term in Eq. 4.16 may be expressed in
vector form for one specific element as
n (v'vc  v(Dvc) + Q) dv = 0
(4.19)
198
where V = solution domain, D = sediment dispersivity tensor, and V =
twodimensional vector operator. The dispersive flux vector F is equal
to DVC. Thus, the second term in the integrand of Eq. 4.19 becomes
NV(DVC) = NVF = V(NF)  (VN) * F (4.20)
The divergence theorem states that
V • (NF) dV = NF • ndS (4.21)
where n = outward normal to the surface S of the domain V. Substituting
Eqs. 4.20 and 4.21 into Eq. 4.19 gives
VN
L_N (V • VC + — • (DVC) + 0) dV  / N DVC • ndS (4.22)
^ N ^
Substituting Eqs. 4.18 and 4.22 into Eq. 4.17 yields for locally
constant dispersion coefficients
A. A
j=NE ac ac ^1 ac
2 //, [N (Q + u— + V—) + (D^ — +
j=i e ax ay ax ^^ax
ac K K
2 N. qj dC = 0 (4.23)
I<=1 ^ "
19?
This equation may be expressed for a singular element by the
element matrix differential equation
a{c}
[k] {C} + [t] + {f} + Cb] {O = 0 (4.24)
at
where
[k] = element steadystate coefficient matrix
[t] = temporal matrix
{C} = vector of unknown nodal concentrations
{f} = element source/sink vector
[b] = boundary or element load matrix.
Equation 4.24 is evaluated for each element with the element load
matrix [b] = 0 for interior elements. The element coefficient, temporal
and boundary matrices are given in Appendix B. The element coefficient
matrix is modified to account for prescribed nodal boundary conditions
by eliminating the row and column corresponding to that nodal unknown.
For those boundary nodes at which no boundary conditions (i.e.
concentrations or fluxes) are prescribed, the normal concentration flux
across that node is set equal to zero.
Next, the element matrix differential equations (Eq. 4.24) are
assembled to form the system matrix differential equation
5{C}
[K] {C} + [T] + {p} + [B] {c} = 0 (4.25)
at
m
where all the matrices and arrays are the system equivalents of those
given in Eq. 4.24.
Rearranging Eq. 4.25 and replacing the partial derivative with a
finite difference gives
[T]
{ + [K] + [B]} {c} + {F} = 0 (4.26)
At
Applying a CrankNicholson type representation to temporally discretize
this equation gives
{—+ 0[K]n+l + 0[B]"'l} {C}"+1 = {—. [(lQ)rK]" 
At At
(ie)[B]"}{c}" + eiF}"""^ + {ie){F}" (4.27)
where 0 = degree of implicitness (e = 1, fully implicit; e = o, fully
explicit), and the superscripts n and n+1 indicate the values of the
arrays and vectors at the current time step (t = nAt) and at the next
time step (t = (n+l)At), respectively. The value of e is specified by
the user. For stability reasons, 9 should be greater than or equal to
0.50. Using the specified initial and boundary conditions, Eq. 4.27 is
solved for the NPNBC unknown nodal concentrations at t = (n+1) At, where
HP = number of nodes in the system and NBC = number of boundary nodes
with specified boundary conditions. The method used to solve Eq. 4.27
is discussed next.
201
4.4.4. Equation Solvers
CSTMH contains two algorithms which solve Eq. 4.27. One algorithm
uses the Gaussian elimination technique and the other uses the frontal
solution program for unsymmetric matrices developed by Hood (1976). The
frontal algorithm was developed specifically for applications of the
finite element method to boundary value problems. Although it is based
on the Gaussian elimination technique, it has advantages over the
conventional banded matrix techniques in that computer storage
requirements and computation times may be considerably reduced in
certain applications. This is especially true for large systems which
might have 1000 or more variables. For small systems (e.g. with 200 or
less nodes), there is no appreciable difference in the computation time
required by the two algorithms. The user of CSTMH specifies whether
the band or frontal method is to be used in solving Eq. 4.27. It is
necessary to create a temporary data file on a disk when the frontal
algorithm is used.
4.5. Convergence and Stability
The accuracy of the numerical scheme used in CSTMH has been
investigated in detail by Ariathurai (1974) and Ariathurai et al.
(1977). These authors reported that rapid convergence to the exact
analytical solutions was achieved for the numerical formulation for the
onedimensional, transient heat conduction problem with and without
radiation, the onedimensional, steadystate and transient convection
diffusion problem, and for the twodimensional Laplace equation.
The results from these convergence tests also indicated that the
combination of the unconditionally stable finite element formulation
202
used to solve the spatial problem and the unconditionally stable Crank
Nicholson type finite difference formulation used to solve the temporal
problem is as well unconditionally stable. However, these tests
revealed that instabilities might still occur when the Peclet numbers
(ratio of advection to dispersion, i.e. ul/d^, where u = flow velocity,
= dispersion coefficient and L = system longitudinal dimension)
become either too large (greater than = 100) or too small (less than 
10' ). For too large Peclet numbers, smaller timesteps must be used to
improve the accuracy of the numerical scheme (Ariathurai et a1 . ,
1977). Too small Peclet numbers rarely ever occur for typical flow
conditions in estuaries, and therefore associated roundoff errors, which
can lead to instabilities, should never be a problem in modeling such
systems. However, spurious results caused by roundoff errors were
encountered in simulating laboratory depositional experiments with CSTM
H. This problem was eliminated by using double precision arithmetic in
the model. No instability problems were encountered in modeling a
prototype system with CSTMH using single precision arithmetic.
A value between 0.5 and 1.0 should be used for the degree of
implicitness in order to insure a stable numerical scheme in time. High
values of this parameter result in a smoother, though no more accurate
solution than values near 0.5. For modeling tidal bodies of water, a
time step of the order 1030 minutes should be used. The element sizes
should be chosen such that the required detail is obtained in critical
areas of concern.
CHAPTER V
MODEL VERIFICATION AND APPLICATION
5.1. Introductory Note
The purpose of this chapter is twofold: 1) to verify the CSTMH
model by demonstrating it's capability of predicting cohesive sediment
transport processes, and 2) to apply the model to a twodimensional,
prototype scale body of water. The first objective is achieved by using
CSTMH to simulate five different laboratory sediment transport
experiments and comparing the measured and predicted results. The
second objective is achieved by using the model to simulate
sedimentation in a coastal marina.
5.2. Laboratory Experiments
A total of four laboratory experiments were conducted at the
Coastal Engineering Laboratory during this study. Three of the four
were conducted in a 18.3 m long, 0.61 m wide and 0.91 m deep
recirculating flume and the fourth in a 0.2 m wide, 0.46 m deep and 0.76
m mean radius rotating annular flume. The experiments in the
recirculating and annular flumes are described in Sections 5.2.1 and
5.2.2, respectively. The results of the model simulations of these
experiments are presented in Section 5.2.3.
203
204
5.2.1. Recirculating Flume Experiments
5. 2.1. a. Facilities
The 18.3 m long flume in which three erosiondeposition experiments
were conducted is shown in Fig. 5.1. A schematic diagram of this flume
is given in Fig. 5.2. The main components of the flume are the
fol lowing:
1) A chamber located at the upstream end of the flume into which water
from the recirculating pipe is discharged. A flow straightening
device was placed in the entry chamber just downstream of the water
discharge pipe.
2) The flume has a steel plated bottom and back wall with glass panels
along the front side. Two electric jacks located 1.5 m downstream
of the entry chamber can be used to tilt the flume to a maximum
slope of 0.02.
3) An underflow tail gate at the downstream end of the flume.
4) A 1.83 m long, 0.91 m wide and 1.2 m deep transition tank, into
which the water in the flume flows.
5) A centrifugal pump with a maximum capacity of 0.164 m^/s.
6) A discharge control valve.
7) A 0.2 m diameter, 15.3 m long PVC return pipe.
8) A 0.33 m wide, 4.9 m long width restricting apparatus was placed in
the upstream half of the flume along the back wall (Fig. 5.1). The
central constant width section was 1.52 m in length, and the two
curved end sections were 1.69 m in length. This apparatus was
constructed out of 0.31 cm plywood. Two coats of fiberglass resin
were applied to the bottom and front and back sides to minimize
swelling of the plywood. Concrete blocks were used for ballast
Fig. 5.1. Downstream View of Recirculation Flume. Width
Reducing Device is Shown on Right Side of Flume.
206
Q. aj
c
>
c
o
c
o
Q, OJ w O
3 a m "w o 9
Q > cr o q: a.
5 ■—
0, O — CM rO T m
U1
a.
a 3
3 to
« OJ o
= 2 o ° S :r: if a
u.Q.a.:SQQ c/jt—
CvJ
CO
s
(T3
E
=3
CD
+>
o
'r—
U
<u
e
Q
y
'I—
u
CVJ
If)
9)
207
(see Fig. 5.1), and silicon and duct tape were placed along the
submerged edges of the apparatus to prevent sediment from seeping
behind and/or underneath the apparatus. This width restricting
apparatus was placed in the flume in order to achieve a region of
higher flow velocities.
A 12 mm diameter, 0.9 m long PVC pipe, capped at both ends, with
1.5 mm diameter holes 5 cm apart, and connected by a rubber hose to
an air compressor, was placed widthwise across the bottom of the
transition tank. The turbulence generated by the jets of
compressed air from this pipe served to increase the turbulence in
the tank. This helped to minimize the possibility of any sediment
depositing in the tank, which had a crosssectional area
approximately three times larger than that of the flume, and
therefore a much lower shear stress.
5.2.1.b. Instrumentation
Velocity measurements were made during the experiments using a Kent
miniflow current meter (Model Number 265). The current meter consists
of a 1 cm diameter, 5 bladed impeller attached to a circular frame,
which is itself connected to a 48 cm long shaft (Fig. 5.3). There is a
0.1 mm clearance between the tip of each blade and the base of the
shaft. Inside the shaft is an insulated gold wire. An electrical
impedance between the gold wire and the shaft is changed by the rotation
of the impeller in a conductive fluid (Wang, 1983). This change in the
impedance modulates a 15kHz carrier signal, the strength of which is
indicated on the needle dial on the monitor box. The variance in the
impedance is a function of the impeller's rotation rate, which is in
208
Fig. 5.3. Kent MiniFlow Current Meter.
Calibration from Supplier
n I I ! \ ! \ 1 ! ! ^ !
^0 2.0 4.0 6.0 8.0 10.0
INDICATOR READING (Hz)
Fig. 5.4. Calibration of Kent MiniFlow Current Meter.
209
turn a function of the fluid's flow velocity. The current meter was
calibrated by Dixit (1982) in the Department of Civil Engineering
Hydraulic Laboratory's flume and by the author in the rotating annular
flume at the Coastal Engineering Laboratory. Figure 5.4 shows the good
agreement obtained between the calibration provided by the manufacturer
and those obtained by Dixit and the author.
The current meter shaft was clamped to a vertically positioned
point gage, which in turn was bolted to a horizontally positioned point
gage (Fig. 5.5). As seen in this figure, the horizontal point gage was
bolted to one side of a wooden cart. This setup permitted exact
(within the 1 mm accuracy of the two verniers) horizontal and vertical
positioning of the current meter inside the flume. The wooden cart was
placed on top of the two 2.5 cm diameter stainless steel rods which
spanned the length of the flume. The cart was moved on top of the rods
to the desired location along the longitudinal axis of the flume.
The timeaveraged elevation of the water surface above the flume
bed was measured during the course of each experiment using the water
surface elevation measuring device shown in Figures 5.6 and 5.7. This
device, which operates on the siphon principle, consists of an electric
point gage, a 5.0 cm diameter plastic tube, two valves, and two 0.5 cm
diameter clear rubber hoses. The point gage was attached to the top of
the tube, as seen in Fig. 5.6, and positioned such that the tip of the
gage touched the water surface at approximately the tube's center. The
effect of surface tension between the inner tube wall and the water
inside the tube did not influence the water surface elevation at the
tube's center because of the relatively large tube diameter (Wang,
1983). A 3.7 mm diameter glass tube was inserted in the unattached end
210
Fig. 5,5. Instrumentation Cart and Setup of Kent
MiniFlow Meter and Two Point Gages.
211
5.7. Setup of Water Surface Elevation Measuring Device
(after Wang, 1983).
212
of the rubber hose. The glass tube was then inserted through a board
located a few inches above the water surface in the flume. This was
done in order to maintain the vertical positioning of the glass tube. A
hand operated suction pump was used to start a siphon between the water
in the tube and in the flume. The slow response time between the water
levels in the flume and tube, caused by the small diameter and the long
length of the connecting hose, resulted in a filtering of the high
frequency, turbulent fluctuations of the water surface. As a result,
only the timeaveraged water surface elevation was measured.
Two of these devices were used to measure the water surface
elevation at three locations along the flume. One device continuously
measured the elevation at a downstream station, while the second device
monitored the elevation at an upstream station and a station in the
reduced width section. As the water surface elevation can be measured
at only one location at a time, the upstream elevation was measured by
first closing the valve of the middle station hose and then opening the
valve of the upstream station hose, and vice versa.
The density profile of the sediment bed in the flume was measured
using a specially designed apparatus (Parchure, 1980). A sketch of this
apparatus 1s given in Fig. 5.8. To obtain a core of the bed the 2 cm
diameter plastic tube is inserted in a vertical position through the
sediment bed until the flume bottom is reached. Then the 15 cm diameter
plexiglass cylinder with a sealed bottom is lowered concentrically
around the plastic tube until it also is positioned on the flume
bottom. The annular space between the 2.5 cm diameter metal tube and
the outer wall of the plexiglass cylinder (see Fig. 5.8a) is filled
with denatured alcohol, to which pieces of dry ice (solid carbon
213
(a) SKETCH OF APPARATUS I
ZZ5 cm
T
1
\
Top Cylinder IScmdia. ^
i
1
i
!
Rastic Tubes d various heights,
0.95 cm dia. glued to Ite
bottom plate
7.5 cm
I
Bottom n
Cylinder P
15cm dia*
Bottom Rate^
(b) SKETCH OF /APPARATUS H
T
15 cm
_L
2 cm dia plastic tube
15 cm dia. plexiglass cylinder
■2.5 cm dia metal tube
■Annular space for mixture of alcohol
and dry ice
Metal
Plate"
Sediment
Porcelein
Dish
Riled with ics ojbes
H7
^Piston with Screw Red
5.8. (a) Apparatus I for Obtaining Sediment Core;
(b) Apparatus II for Sectioning a Frozen Sediment Core
(after Parchure, 1980).
214
dioxide) are added. The combination of the alcohol and dry ice causes
the sediment core inside the plastic tube to freeze in approximately 20
minutes. The plastic tube is then removed and placed horizontally in a
second apparatus, as shown in Fig. 5.8b. The rectangular box is filled
with ice cubes in order to keep the sediment core frozen. The piston
with threaded rod is used to push approximately 5 mm lengths of the
frozen core out of the plastic tube at a time. The ejected section of
the core is brought in contact with a metal plate in front of the core
(see Fig. 5.8b), which causes this 5 mm section of the core to quickly
melt and drop into the porcelein dish. This process is repeated until
the entire sediment core has been collected in separate dishes. The
sediment 1n each dish is oven dried and weighed in order to determine
the dry sediment mass in each 5 mm section of the frozen core. The
method used to determine the density profile is described in the next
section.
Samples of the sedimentwater mixture were collected at periodic
intervals throughout the experiments using the water sampling device
shown in Fig. 5.9. The horizontal sampling tubes have a 3.2 mm diameter
and are spaced 4.0 cm apart. This device works on the siphon principle,
so that when a sample is desired, the end of the rubber hose is lowered
below the level of the horizontal sampling tube in the flume and
approximately a 60 ml sample of the suspension is collected in a sample
bottle. The volume of water contained in the hose is drained into
another container, and then the 60 ml sample is collected in the
bottle. This procedure was used so that a sample of the suspension
which existed in the flume at the sampling time was collected, and not
that at the previous sampling time. In all the experiments, two samples
were collected simultaneously from the two lowest sampling tubes.
215
Fig. 5.9. Water Sampling Device.
GRAIN SIZE (mm )
Fig. 5.10. Grain Size Distribution of Kaolinite Used for the
Experiments.
216
The suspension concentration of each water sample was determined
using a Millipore filtering apparatus, an oven, and a Mettler balance
{Model No. H80) with a ±0.05 mg accuracy. The following procedure was
used:
1) Withdraw a certain volume of the suspension from the sample bottle
using a 10 ml pipette.
2) Filter this sample through a preweighed Millipore filter paper
with a 0.45 m pore diameter.
3) Place the filter paper containing sediment in an oven at 50 °C for
at least two hours.
4) Weigh the dried filter paper.
5) Calculate the suspension concentration C = (mass of dried filter
paper containing sediment  mass of filter paper alone) / (volume
of suspension filtered).
Commercial grade kaolinite was the sediment used in the three
experiments performed in the recirculating flume. The particle size
distribution of the kaolinite, determined by a standard hydrometer test,
is shown in Fig. 5.10. The median particle diameter, as seen in Fig.
5.10, was 1 \im. The CEC value for the kaolinite given by the suppliers,
the Feldspar Corporation, Edgar, Florida, was 5.26.5 meq / lOOgm
(Dixit. 1982). Tap water was used as the fluid in these experiments.
The chemical composition of this water is given in Table 5.1. A two
week period was used to equilibrate the tap water and about 75 kg of
kaolinite before the first experiment was conducted.
217
Table 5.1
Chemical Composition of the Tap Water (after Dixit, 1982)
CI
26
ppm
N03
0.07
ppm
Fe
0.5
ppm
K
1.4
ppm
Ca
25
ppm
Mg
16
ppm
Na
10
ppm
Total Salts
278
ppm
pH
8.5
5.2. I.e. Procedure
A zero bed slope was used in all three experiments. The water
surface elevation measuring devices at the upstream and downstream
stations were used to adjust the slope to zero.
The following procedure was used to form a flowdeposited sediment
bed in each of the experiments. The flume pump was started and the 75
(dry mass) of sediment was mixed at a shear stress of approximately 0.5
N/m for four hours. The flume calibration performed by Dixit (1982)
was used to estimate the shear stress. The mixing was artificially
enhanced by pushing a rubber wedge along the bottom of both the flume
and transition tank in order to initially suspend all the sediment.
After four hours of mixing, the flow in the flume was reduced to a
shear stress of approximately 0.025 N/m^, which was maintained for eight
218
hours. Most of the sediment deposited during this deposition period.
The nonuniform flow in the flume, which was caused by the width
reducing device and the flow under the downstream tail gate, resulted in
a bed of variable thickness along the length of the flume.
After the eight hour deposition period, the flow was stopped
completely. The flow deoosited bed was allowed to undergo selfweight
consolidation for the following consolidation times: J^^ = 3, 84 and
240 hours for Test No. 1, Test No. 2 and Test No. 3, respectively.
At the end of the consolidation period, the flume pump was started
and the flow rate was slowly adjusted (i.e. increased) so that after 15
minutes the shear stress in the flume was approximately 0.02 N/m^. This
same flow rate was maintained for an additional 45 minutes. This
procedure was followed in order to resuspend the sediment which
deposited during the consolidation period in the recirculating pipe,
which has a flow crosssection approximately four times smaller than the
flume and therefore a shear stress approximately 16 times greater than
that in the flume, without resuspending any of the sediment in the
flume. The suspension concentration which existed in the flume after
this one hour period was regarded as the initial concentration, Cq, at
the start of each experiment. was determined by water samples
collected at the end of the one hour resuspension period. In addition,
the bed surface elevation which existed at the end of the one hour
period was measured through the front glass panels every 0.91 m along
the flume test section. These measurements were used as initial bed
conditions in the model simulations.
The approximate shear stress history for the three experiments are
shown in Fig. 5.11. This figure may be interpreted as follows. In Test
219
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^b2
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(c) TEST NO. 3
80
ICO
Fig. 5.11. Shear Stress History for Experiments in the Recirculating
Flume.
220
^ ^'''dc ^ hours), the approximate shear stress in the unrestricted
width sections of the flume was equal to 0.06 N/m^ for the first two
hours, then it was increased to 0.12 N/m^ for the third and fourth
hours, and finally it was decreased to 0.033 N/m^ for the final five
hours. In Test No. 2 (T^^ = 84 hours), the approximate shear stress was
0.075 N/m^ for the first 3.5 hours, 0.17 N/m^ for the next 2.5 hours,
and 0.026 N/m^ for the final five hours. In Test No. 3 (T^^ = 240
hours), the approximate shear stress was 0.17 N/m^ for the first two
hours, 0.026 N/m^ for the next five hours, and 0.075 N/m^ for the final
two hours. It took approximately two minutes to change the flow rate in
the flume to the new shear stress value and establish steady flow
conditions.
During each of the constant shear stress time intervals, water
samples were collected at sampling Station A (see Fig. 5.12) from the
two lowest sampling tubes at 0, 2, 5, 10, 20, 30, 40, 50, 60, 75, 90,
105, 120 minutes, etc. after the shear stress was changed. In order to
determine if any spatial (i.e. longitudinal and transverse) variability
in the suspension concentration occurred during the experiments, water
samples were also taken at alternate sampling times at Station B during
Test No. 2 and at Station C during Test No. 3. Figure 5.12 shows the
reach of the flume used in each experiment, and the locations of the
three water sampling stations (A, B, C), the three water surface
elevation measuring stations (D, E, F) and the three velocity measuring
stations (G, H, J).
The water surface elevation was monitored periodically at each of
the three measuring stations during each constant shear stress time
interval. The bed thickness was measured every 0.91 m along the length
221
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222
of the flume's test reach at least once during each constant shear
stress time interval. In addition, the water temperature was measured
every hour over the duration of each experiment.
The vertical velocity profiles at five lateral positions at
Stations G and J, and at three lateral positions at Station H (see Fig.
5.12) were measured once during each constant shear stress interval
using the Kent mini flow meter.
After the three experiments were completed, the identical bed
formation process was again repeated. A frozen core of the sediment was
obtained for the purpose of determining the density profile at each of
the consolidation times (i.e. 3, 84 and 240 hours) used in the
experiments. These cores were collected downstream of the width
restricting device, where the bed had approximately uniform thickness.
The following procedure was used to determine the bed density
profile from each segmented core. The density of each 5 mm thick layer
was determined by dividing the dry sediment mass in that layer by the
volume. Freezing of the core caused the sediment sample inside the core
to swell. A uniform correction for the swelling was made by dividing
the total thickness of the frozen core by the total thickness of the
original (i.e. before freezing) core, and then dividing the thickness of
each frozen layer (=5 mm) by this factor. This gives the mean density
in each layer, from which the density profile may be constructed.
5.2. l.d. Results
Figures 5.135.15 show the variation of the suspension
concentration with time during Tests No. 1, 2 and 3, respectively. The
concentrations determined for both the lower and upper sampling tubes
223
0.0 2.0 4.0 6.0 8.0
TIME (hrs)
Fig. 5.13. Measured and Predicted Suspended Sediment Concentrations
for Test No. 1.
224
00' I 1 I I I I I I I I
0.0 2.0 40 6.0 8.0 10.0
TIME (hrs)
Fig. 5.14. Measured and Predicted Suspended Sediment Concentrations
for Test No. 2.
225
6.0
I 4.0
tr
H
UJ
u
z
o
u
o
CO
UJ
a.
3
C/)
2.0
Station
• A Upper
^ A  Lower
o C Upper
^ C  Lower
n 1 r
Test No. 3
V 240 hrs
Concentrations
— Predicted
X
o
Measured
'•A o ^
00
0.0
2.0
4.0
TIME (hrs)
6.0
8.0
Fig. 5.15. Measured and Predicted Suspended Sediment Concentrations
for Test No. 3.
226
are shown in these figures. In addition. Fig. 5.14 shows the suspension
concentrations determined for the downstream collection station (Station
B) during Test No. 2, and Fig. 5.15 shows the measured concentrations at
Station C during Test No. 3. As evident in these two figures, the
suspension concentration varied slightly in the flume in both the
longitudinal and lateral directions. The lateral gradient in C was
caused by the nonuniform lateral velocity profile which existed in the
flume. The longitudinal gradient was caused by the combined effects of
longitudinal dispersion, erosion and/or deposition which occurred along
the length of the flume. The concentration values determined from the
lower sampling tubes were plotted in Figs. 5.135.15 only when they
differed from the values at the upper sampling tubes by more than ±0.03
g/1.
No significant difference (i.e. vertical gradient) in the
suspension concentration was observed between the upper and lower
sampling locations during the three experiments, except for the time
period immediately following changes in the flow rate. This phenomenon
is best exemplified by the large vertical variation in C (up to 2 g/1)
during the first half hour period of deposition (from 6.0 to 6.5 hours)
in Test No. 2 (see Fig. 5.14). Differential settling is the probable
cause for this observed vertical concentration gradient during the
initial stages of deposition. The high vertical variation in C (up to
approximately 1 g/1) found, for example, during the first half hour
period of erosion (from 2.0 to 2.5 hours) in Test No. 1 (see Fig. 5.13)
is probably due to higher resuspension rates than vertical diffusion
rates during this initial period. As the resuspension rate decreases
due to increasingly smaller values of the excess bed shear (caused by
227
the Increase of the bed shear strength with depth below the Initial bed
surface), the continuing vertical diffusion of sediment reduces the
magnitude of this vertical concentration gradient, as seen in Fig. 5.13
from 2.5 to 4.0 hours.
The depthaveraged velocity at each vertical was determined by
plotting the vertical velocity profile and then integrating it using a
planimeter. The water depths at Stations D, E and F were determined by
subtracting the measured bed thickness from the water surface elevation
above the bottom of the flume.
The bed density profiles for the three experiments are given in
Fig. 5.16. The applicability of the power law relationship between the
dry sediment density, p, and the depth below the bed surface, z^, as
given by Eq. 3.76, is apparent for the three measured density profiles.
5.2.2. Rotating Annular Flume Experiment
The purpose for conducting an experiment in the rotating flume at
the Coastal Engineering Laboratory was to verify that the cohesive
sediment transport model developed during this study can be used to
predict erosion and deposition rates in an unsteady flow field.
5. 2. 2. a. Facilities
The annular flume in which the experiment was performed is shown in
Fig. 5.17. This flume has the following dimensions: 0.21 m wide, 0.45
m deep and 0.76 m mean radius. The flume consists of three main
components: 1) a rotating circular fiberglass channel which holds the
sedimentwater mixture, 2) an annular ring with a slightly smaller width
and the same mean radius as the channel, and 3) a steel frame and
228
p / p
. 5.15. Measured Bed Density Profiles for Experiments in
Recirculating Flume.
229
230
electric motors. The ring, positioned in contact with the water
surface, and the channel are rotated simultaneously in opposite
directions in order to achieve a nearly uniform turbulent shear field in
the channel, and to minimize the effects of rotationinduced radial
secondary currents. This design and operational procedure eliminates
the need for aggregatedisrupting elements such as circulatory pumps, in
which very high shearing rates usually occur. The required bed shear
stress is attained by adjustment of the rotational speeds of the channel
and the ring. Four taps, located on the outer channel wall (see Fig.
5.17), are used to collect suspended sediment samples from the channel.
5.2.2.b. Instrumentation
A Hewlett Packard HP85 microcomputer with two digitaltoanalog
converter units were used to control the rotation rates and
accelerations of the channel and the ring. The microcomputer was
programmed to generate the desired flow field in the channel.
The bottom sediment from Lake Francis, Nebraska and tap water with
a 10 ppt solution of commercial grade sodium chloride were used in this
experiment. Xray diffraction analysis performed by the Soil
Characterization Laboratory at the University of Florida revealed that
this sediment is predominantly composed of montmorillonite, illite,
kaolinite and quartz.
5.2.2.C. Procedure
The sediment and 10 ppt saline water were placed in the channel. A
30.5 cm depth of flow was used. The sedimentwater mixture was mixed at
a shear stress of approximately 1.7 N/m^ for 24 hours. The flume was
231
then stopped and the suspended sediment was allowed to deposit and
undergo selfweight consolidation for 40 hours.
The HP85 microcomputer was programmed to generate a uni
directional, semidiurnal, constant depth tidal flow. The cross
sectionally averaged sinusoidal velocitytime record used in this tidal
cycle experiment is given in Fig. 5.18. Because of a mechanical probl
with the bearings in the channel's drive shaft, it was not possible to
have a true slack period in this experiment. Though not apparent in
Fig. 5.18, the flow velocity was discretized into five minute
increments, during which the velocity varied as follows: during the
first 30 seconds of each five minute time increment, the velocity was
linearly increased or decreased to the next value (as determined by a
sinusoidal velocitytime relationship), while for the remaining 4.5
minutes, a constant velocity was maintained.
The program was started and run for two tidal cycles (25 hours) in
order to establish quasi steady state conditions. Water samples were
collected from the middepth tap at the end of this 25 hours and every
five minutes thereafter for 16 hours. The samples were collected just
before the end of each five minute increment (i.e. before the velocity
was changed).
The temperature of the water in the channel was measured every
hour. In addition, water was added every hour to replace that withdrawn
in the water samples in order to keep the ring in contact with the water
surface.
em
232
233
5.2.2. d. Results
Figure 5.18 shows the variation of the suspension concentration
over the duration of the 16 hour experiment. Also plotted in this
figure is the variation of the crosssectionally averaged velocity. A
short lag between the flow velocity and the suspension concentration is
seen to occur immediately following the occurrences of minimum and
maximum velocities.
5.2.3. Model Simulations
The test reach of the recirculating flume was divided into the 34
element, 141 node finite element grid shown in Fig. 5.19. Zero sediment
flux boundary conditions were used for the upstream and downstream flow
boundaries. The depthaveraged velocities measured at Stations G, H and
J were used for the nodal velocities in elements 12, 710, and 1534,
respectively. The nodal velocity vectors in the converging section
(elements 36) of the flume and in the diverging section (elements 11
14) were determined from continuity considerations. Likewise, the water
surface elevations measured at Stations D, E and F were used for the
nodal values in elements 12, 710, and 1534, respectively. The nodal
water surface elevations in elements 36 and 1114 were determined by
linear interpolation of the values measured at Stations D and E, and
Stations E and F, respectively. The initial bed thicknesses measured
every 0.91 m along the flume were used as the initial bed thicknesses
for the evennumbered elements. The initial bed thickness of the ith_
oddnumbered element was assumed to be equal to that of the (i+l)th
evennumbered element, for i=l, 33. The dry mass of the sediment
forming the bed in each element at the start of each experiment was
234
(r
5^
6
7
8
9
10
II /
12
/,4
15
16
17
IB
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Fig. 5.19. Finite Element Grid of Recirculating Flume; Distorted
Sketch  Width: Length = 4.1: 1.0.
235
determined using the measured bed thickness and measured bed density
profile. The dry mass in each element and the measured bed density
profile were read into CSTMH in order to form the initial bed for each
experiment. The parameters which characterize the erosional and
depositional characteristics of kaolinite in tap water, determined by
Dixit (1982) and Mehta (1973), respectively, and the consolidation
parameters given in Chapter III, Section 3.7.1 were used in CSTMH to
simulate the three experiments in the recirculating flume. The
dispersion coefficients were calculated using the dispersion
algorithm. A two minute time step was used in the model simulations.
Comparisons of the predicted and measured suspension concentrations
for the three experiments are shown in Figs. 5.135.15. Satisfactory to
good agreement is observed in all three experiments. A discussion of
these results is given in Section 5.4.
A four element, 23 node straight grid was used to represent the
annular flume in simulating the tidal cycle experiment. The length of
this grid system was set equal to the circumference of a circle with a
radius equal to the mean radius of the flume. The suspension
concentration at the "downstream" flow boundary was used for the
"upstream" boundary condition in order to represent they were the same
boundary. The velocity record shown in Fig. 5.18 and the erosional and
depositional characteristics of Lake Francis sediment reported by Mehta
et al_. (1982a) and Hayter and Mehta (1982) were used in CSTMH. A 2.5
minute time step was used in the model simulation.
A comparison of the predicted and measured suspension
concentrations is as well shown in Fig. 5.18. A discussion of the
simulation result is given in Section 5.4..
236
5.3. Simulation of WES Deposition Experiment
A deposition experiment conducted in the 99.7 m long, 0.46 m deep
and 0.23 m wide nonrecirculating flume at the U.S. Army Corps of
Engineers Waterways Experiment Station, Vicksburg, Mississippi was
simulated with CSTMH. A schematic diagram of this flume is shown in
Fig. 5.20. A detailed description of the flume is given by Dixit etal.
(1982). The purpose of this experiment and three other experiments
conducted in the 99.7 m long flume was to investigate the phenomenon of
sediment sorting in the longitudinal direction, the occurrence of which
is well documented in muddy estuaries (Edzwald et al_. , 1974; Dixit et
al_'. 1982).
The sediment used in this experiment was the commercial grade
kaolinite described in Chapter III. The fluid used was tap water with a
chloride concentration of 18 ppm. pH = 7.8 and sodium adsorption ratio
SAR = 2.07 (Dixit etal., 1982).
The procedure used in this deposition experiment was the
following. A 100 g/1 sedimentwater slurry was prepared by mixing the
kaolinite and water for three hours. The slurry was injected at a
specified rate into tap water which flowed from the headbay into the
flume (see Fig. 5.20). The slurry injection rate was regulated so that
the resulting suspension had a concentration of approximately 4.0 g/1.
The flow rate was held constant for three hours, after which only the
clear water flow was maintained until the sediment cloud had passed out
of the flume. The thickness of the deposited bed was measured 15
minutes after the end of the experiment. Vertical and horizontal
velocity profiles and the water surface elevations were measured at
several stations along the length of the flume. In addition, vertical
237
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238
suspension concentration profiles were measured at several stations
along the centerline of the flume every 15 minutes during the course of
the experiment.
The mean flow depth, mean velocity and the bed shear stress during
the experiment were 0.162 m, 0.091 m/s, and 0.033 N/m^, respectively
(Dixit etal_., 1982).
This experiment was simulated using a 10 element, 53 node grid and
a four minute time step. The stated flow conditions, the depositional
properties of kaolinite in tap water found by Mehta (1973) and the
consolidation properties of kaolinite given in Chapter III, Section
3.7.2 were used in CSTMH to perform the simulation. For the upstream
boundary condition, the suspension concentration was set equal to 4.0
g/1 for three hours, and then set to 0.0 g/1 for the remainder of the
experiment. A zero concentration flux was used for the downstream
boundary condition. The model simulation was continued until the
maximum predicted concentration in the flume was 0.04 g/1 (1% of the
initial injection concentration). Figure 5.21 shows the comparison
between the measured and predicted bed thickness profile at the end of
the experiment. A discussion of the results of this simulation is given
in the next section.
5.4. Discussion of Results
The model simulations of the three experiments in the recirculating
flume yielded good agreement between the measured and predicted temporal
variations in the suspended sediment concentration, as seen in Figs.
5.135.15. In Test No. 1, it is apparent from the measured
concentrationtime record that during the two erosion intervals (i.e.
239
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(ujo) SS3N>iDIHl 039
240
during the first four hours) the bed was primarily redispersed (mass
eroded) by the flowinduced bed shear stress. Deviations between the
measured and the predicted suspension concentrations (i.e. between the
observed and the predicted rates of erosion) occurred because the
consolidation algorithm in CSTMH stipulated that the bed, for which T^^
> 2 hours, would undergo the slower process of resuspension when
subjected to an excess shear. Even though the predicted erosion rate
during the first hour of each erosion interval was less than that
reflected in Fig. 5.13, good agreement was achieved between the total
mass of sediment that was measured and predicted to erode over the
duration of each interval. Nevertheless, the fact that the bed was
apparently redispersed when subjected to the given bed shear stress
points out a limitation of the erosionconsolidation algorithm in CSTMH
and an area for future research.
The deposition stage in Test No. 1 was in Range IB (see Fig. 3.48)
where the settling velocity decreases with a decrease in the suspension
concentration according to W3 a c\ with n = 1.33. The good agreement
obtained between the measured and the predicted concentrations in this
interval (see Fig. 5.13) was obtained by adjusting the value of the
proportionality constant (kJ in Eq. 3.51b) between and c".
In Test No. 2 with T^^ = 84 hours, the measured suspended sediment
concentrations again indicate that the top layer of the bed was
apparently redispersed during the first erosion step (see Fig. 5.14). A
slightly better agreement between the measured and the predicted
concentration was achieved during this first step by increasing the
value of the parameter a (see Eq. 3.13) in the top bed layer from the
value obtained by Dixit (1982).
241
The measured concentrations during the second erosion interval in
Test No. 2 indicate that the bed eroded aggregate by aggregate
(resuspended) after the initial five minute interval (from 3.50 hours to
3.58 hours), during which redispersion occurred. Because the bed was
resuspended during all but the first five minutes of this interval, a
better agreement was obtained between the measured and the predicted
concentrations, and therefore also between the actual and the predicted
rates of erosion. However, as apparent in Fig. 5.14, the predicted
concentrations were consistently lower than the measured concentrations
during the second erosion interval. A better agreement could have been
obtained by increasing the values of the aggregate erosion rate (see
Eq. 3.13).
Good agreement was again achieved during the deposition stage in
Test No. 2, although the predicted deposition rate was slightly higher
than that observed during the last two hours of the experiment. This
resulted in a lower predicted than observed concentration at the end of
the experiment.
In Test No. 3 with T^^ = 240 hours, resuspension occurred during
the first erosion interval after the first 10 minutes during which
redispersion apparently occurred. The redispersion resulted in higher
observed than predicted suspension concentrations for the first hour.
Fairly good agreement was achieved during the second hour.
Good agreement was obtained between the measured and the predicted
concentrations during the five hour period of deposition in Test No. 3
(see Fig. 5.15), although from 2.5 hours to 5.0 hours the predicted
deposition rates were slightly less than the observed rates, while
during the last hour of the deposition interval (from 6.0 hours to 7.0
242
hours) the predicted deposition rates were slightly greater than the
observed rates.
The measured suspension concentrations during the second erosion
step in Test No. 3 (from 7.0 hours to 9.0 hours) indicate that the top
portion of the sediment which deposited during the previous five hours
was eroded in mass by the increased bed shear stress. Redispersion was
simulated by CSTMH to occur during the first time step of increased bed
shear. The occurrence of redispersion is not evident in Fig. 515
because of the extremely thin unconsolidated new deposit (UND) layer
(and therefore small quantity of dry sediment mass which forms this
layer) used in this simulation. The subsequent resuspension predicted
by CSTMH yields a slightly greater eroded sediment mass than measured.
Figure 5.18 shows the comparison between the model simulation of
the tidal cycle experiment in the rotating flume and the measured
suspension concentrations. A good agreement between the predicted and
measured concentrations is seen. The most noteworthy differences are
that the measured concentrations lag the predicted values by
approximately 20 minutes at the times of maximum concentrations (which
correspond to the times of maximum or peak tidal flow velocities), while
the predicted concentrations lag the measured values by approximately 20
minutes at the times of minimum concentrations (which corresponded to
the times of minimum flow velocities). Thus, assuming the velocitytime
record used in the simulation is correct, the predicted periods of
erosion and deposition are approximately 40 minutes longer and shorter
than, respectively, the observed periods. There are at least four
possible explanations for this difference in the predicted and observed
periods of erosion and deposition. 1) A time lag between the change in
243
the rotation rates of the ring and the channel of the rotating annular
flume and the resulting response in the flow velocity. 2) The
occurrence of deposition even in an accelerating flow when the flow
velocity is below a certain minimum critical value. 3) The decrease in
the flow acceleration leading up to a maximum velocity might cause a
portion of the suspended sediment in the upper half of the water column
to settle below the middepth tap where the water samples were
obtained. This would result in a decrease in the suspension
concentration as determined from the middepth sampling location, and
therefore seem to indicate that deposition had occurred, even though the
sediment might have just settled below middepth and not have actually
deposited on the bed. 4) The 20 minute lag between the onset of
accelerating flow and the increase in the measured suspension
concentration (as observed at about 6.0 hours in Fig. 5.18) might have
been caused by the lag time between the erosion of sediment and the
vertical diffusion of this sediment up to the middepth sampling
location. The merits of the second and fourth possible explanations are
questionable in light of the fact that no time lag is observed between
the measured and predicted concentrations at 12.5 hours in Fig. 5.18.
The first possible explanation is not very plausible since the response
time of the water in the channel to changes in the rotation rates of the
ring and the channel is generally of the order of one to three
minutes. The explanation given (number three) for the observed lag
between the predicted and measured concentrations at maximum flow
velocities needs to be investigated in the future by repeating this
experiment and taking water samples at several locations over the flow
depth during the latter half of accelerating flows.
244
As observed in Fig. 5.18, the measured increase in the amount of
sediment eroded per half tidal cycle (i.e. during the two accelerating
flow periods per tidal cycle) was predicted fairly accurately by CSTM
H. The explanation for this slight increase in eroded sediment is the
following. After the top bed layer is eroded, the shear strength of the
now exposed lower bed layer decreases as the bed surface swells in
response to the removal of the overburden pressure. This phenomenon is
simulated in CSTMH by changing (decreasing) the bed shear strength at
the new bed surface to the value of that existed at the bed surface
at the end of the previous time step. Thus, as the experiment
continues, the bed shear strength at the various depths (below the
initial bed surface) to which the bed is eroded becomes slightly less,
which of course increases the susceptibility of the exposed sediment to
erosion. This slight decrease of the bed shear strength of the surface
due to swelling is greater, on a short term basis, than the increase of
the shear strength due to consolidation. The fact that CSTMH simulates
the slight increase in the amount of sediment eroded per time step with
good accuracy, as seen in Fig. 5.18, is an indication that this
representation of the decrease in i^ at the bed surface is realistic.
The simulations of the three experiments in the recirculating flume
and tidal cycle experiment in the rotating flume have verified the
predictive capability of CSTMH. The ability of CSTMH to model the
longitudinal sorting process which occurred in the deposition experiment
in the flume at WES is discussed next.
As apparent in Fig. 5.21, satisfactory agreement was achieved
between the measured and predicted deposit thicknesses in the lower 65 m
reach of the WES flume. The measure thickness in the first 35 m
245
possibly shows the influence of longitudinal sorting, which results in a
variation in the rates of deposition of the suspended aggregates along
the flume. A causative factor of this phenomenon is thought to be
differences in the composition of the particles which form the
aggregates (Dixit etal_., 1982). CSTMH is not capable of simulating
longitudinal sorting because the parameters (specifically tgQ and
which characterize the lognormal depositional law are assumed to be
spatially invarient. However, if the relationships between t^Q and
with distance along the flume were known, it would be possible to
incorporate these into the deposition algorithm, and thereby have the
capability of predicting the effect of longitudinal sorting on the rates
of deposition. When CSTMH is used to predict the sediment movement in
an estuary, the variation of tgg and in both the x and y directions
would have to be determined. The number of field and laboratory
experiments that would need to be performed in order to determine the
relationships tgQ = tgg (x, y, t^) and = a, (x, y, , (or even tgQ
" ■'^50 '^b^ ^2 " °2 "^b^^ ''^ thought to be impractical. Thus,
only the relationships t^Q = t^Q [x^) and = (t^) were incorporated
into CSTMH.
Another possible explanation for the smaller measured bed thickness
in the upstream 35 m reach is the increased turbulence present in the
flume immediately downstream of the point of injection. Increased
turbulence would result in lower rates of deposition, and therefore to
smaller bed thicknesses. It needs to be emphasized that the preceding
discussion on the discrepancy in the upstream 35 m reach is based on the
assumption that the bed thickness in this reach is approximately uniform
and equal to the measured value at 25 m.
246
5.5. Model Applications
The utility of CSTMH is demonstrated by simulations of the
sedimentation in Camachee Cove Marina and the suspended sediment
transport in a 10 km hypothetical canal. Camachee Cove Marina is
located on the Tolomato Ri verIntercoastal Waterway about 150 m north of
the Vilano Bridge in St. Augustine. Florida. An aerial photograph of
the basin is shown in Fig. 5.22. The single entrance channel to the
basin is about 180 m long and 60 m wide, and has naturally sloped
banks. The surface area of the basin is approximately 33.370 m^. The
semirectangular shaped basin has approximate dimensions of 300 m in
length and 100 m in width, being tapered towards the south end
(Srivastava. 1983). The bathymetry of the marina basin determined from
a survey conducted in September. 1982 is shown in Fig. 5.23. The local
scour holes seen in this figure were probably caused by boat
propellers. The bulkhead of the basin is made of concrete sheetpile.
The sediment material in the basin was found to have eight percent of
organic matter by weight, and a median particle diameter which varied
from 10 m in the northern end of the basin to 40 m in the center of
the basin. Such a distribution is not surprising since the largest
particles would be expected to deposit in the central wide section of
the marina.
The hydrographic and sediment data required to model both the
predominantly tideinduced circulation and sedimentation in the basin
were collected by the Coastal Engineering Laboratory at the University
of Florida. A detailed description of the field study is given by
Srivastava (1983). The main findings of this hydrographic study were
the following: 1) The tide is semidiurnal, with a mean range of 1.4
247
Fig. 5.23. Bathymetry of the Entire Basin Obtained in September, 1982.
249
m. 2) The maxlinum velocity in the entrance is approximately 2.4
cm/s. 3) The average suspended sediment concentration was 6.2 tng/1 for
the months of July and August, 1982. and 6.8 mg/1 for November, 1982
through July, 1983.
The hydrodynamic modeling was performed using the twodimensional,
depthaveraged finite element flow model RMA2 (Norton et al .. 1973).
The finite element grid used in modeling both the tidal flow and
cohesive sediment transport is shown in Fig. 5.24. The results from the
flow modeling as well as the required sediment data (e.g. suspension
concentrationtime record at the entrance, sediment settling
characteristics) given by Srivastava (1983) were used in modeling the
sedimentation in the marina using CSTMH. The results from this
modeling effort are shown in Fig. 5.25, which shows contours of the
predicted amount (thickness) of sediment deposition in centimeters per
year. The mean depth of the marina is predicted to reduce by 8.6
cm/year, which is 42% less than the measured 14.8 cm/year (Srivastava,
1983). The measured sedimentation rate was obtained by comparing
bathymetric surveys conducted in March, 1980 and September, 1982. The
observed deposition pattern is not unexpected, as the greatest amount of
sediment deposition would be expected in the central wide portion of the
basin where the flow velocities are considerably lower than in the
relatively narrow channel. Three limitations of this modeling effort,
and the probable reasons for the difference between the measured and
predicted deposit thickness, are discussed next.
The predicted sedimentation rate is representative of fair weather
conditions only, in that the tide and suspended sediment concentration
at the marina entrance (which were used for the boundary conditions in
250
251
252
the flow and sediment transport model, respectively) were measured
during fair weather. It is believed that storm events may enhance the
rate of sediment intrusion into the basin. This is attributable to the
probable increase in concentration of suspended sediments caused by the
erosion of more sediment by the storm agitated exterior body of water
(i.e. the Tolomato River).
The water samples collected over the depth of flow in the entrance
channel during the field study, from which the depthaveraged suspension
concentrations were determined, were only collected down to 30 cm above
the bottom. Thus, the proportion of the suspended load, which may be
considerable, that is transported into and out of the basin in the
bottom 10% of the flow depth was not accounted for in the boundary
conditions used in the cohesive sediment transport model.
The influence of horizontal circulation due to shear flow at the
entrance to the marina, as well as that due to windinduced vertical
circulation was not accounted for in the flow modeling. Thus, only the
advective and dispersive transport of suspended sediment due to tidal
flow in the marina was accounted for in the sediment transport
modeling. The combination of these three factors, which were not
accounted for in this modeling effort, are felt to be the reason why the
predicted sedimentation rate was 42% less than the measured rate.
The dimensions of a 10 km long hypothetical canal are shown in Fig.
5.26. The canal was divided into nine elements and 48 nodes, with the
length of elements 1  3 equal to 833 m and that of elements 4  9 equal
to 1250 m. The canal was assumed to have a uniform bottom roughness, as
quantified by a Mannings coefficient of 0.02, which is a reasonable
value for a straight natural waterway with a muddy bottom. The depth
254
and the mean velocity at nodes 1. 2 and 3 were taken to be 5.0 m and 0.5
m/s respectively. The velocities and water depths at nodes 4  48 were
evaluated using the conservation of energy and mass equations for an
open channel. The total drop in the water depth over the 10 km distance
due to frictional resistance and the gradual enlargement in width at
element 5 was determined to be 0.16 m. The initial suspension
concentration in the canal was taken to be 0.0 g/1. The following
boundary conditions were used: nodes 1. 2 and 3: C(t) = 0, and nodes
46, 47 and 48: 9c{t)/ax = 0. The upstream (i.e. nodes 1. 2 and 3)
boundary condition states that no suspended sediment was transported
into the canal from upstream sources, while the downstream (i.e. nodes
46, 47 and 48) boundary condition stipulates that the longitudinal flux
of suspended sediment across the downstream boundary was zero. In
elements 1  4. an initial, partially consolidated Lake Francis sediment
bed 0.17 m in thickness was assumed to exist, while in elements 5  9,
no initial bed was present.
Erosion of the initial sediment bed occurred in elements 1  4,
while deposition of the sediment suspended in the first four elements
occurred in elements 5  9. The suspension concentrationtime record
for elements 4 and 5 are shown in Figs. 5.27 and 5.28 for salinities of
0, 1, 10 and 35 ppt. As evidenced by the over threefold decrease in
concentration between Figs. 5.27 and 5.28, a high percentage of the
suspended sediment deposited in element 5. Also observed in these two
figures is a reduction in the quality of sediment suspended with
increase in the salinity. This observation follows from the previously
described effect of salinity on the rate of erosion. Also apparent is
the small effect of salinity on the rates of deposition. In conclusion.
255
Fig. 5.27. Predicted Suspended Sediment Concentrationtime Record
for Element 4 in Hypothetical Canal.
256
TIME (Hrs)
5.28. Predicted Suspended Sediment Concentrationtime Record
for Element 5 in Hypothetical Canal.
257
these simulations demonstrate the significant influence of salinity on
the transport rate of cohesive sediments for salinities less than about
10 ppt, and the diminished influence of salinity for salinities greater
than 10 ppt.
5.6. Model Limitations
A twodimensional, depthaveraged model such as CSTMH can strictly
be applied only to estuaries, harbors and basins (such as marinas) where
the horizontal dimensions of the water body are at least one order of
magnitude greater than the vertical dimension. Applications to
partially mixed water bodies or especially to highly stratified water
bodies should be made when only rough estimates of some sedimentary
process (e.g. shoaling rate) are required.
Currently the model has the capability of simulating the movement
of only one constituent (e.g. cohesive sediment, water temperature, or
algae, provided the source/sink expressions for a nonconservati ve
constituent are known). It is possible, however, to modify the model so
that any number of constituents may be incorporated.
Probably the main "limitation" of a model arises from three
sources: 1) insufficient data, 2) poor quality of data and 3)
limitations of the hydrodynamic modeling. The first two sources are
attributable to the fact that, owing mainly to time and cost
consideratins, all the bathymetric, hydraulic and sedimentary data
required for use in such a model are rarely, if ever, measured and/or
collected in the body of water being modeled. In addition, the quality
of the data is often questionable. Data requirements and the field
collection and laboratory testing programs required to obtain these data
are briefly described in Appendix D.
258
The third source is itself often the result of the first two,
inasmuch as progress has been achieved in the past two decades in
modeling estuarial hydrodynamics (Leendertse et al_. , 1973; King et al.,
1973; Liu and Leendertse, 1978).
The importance of experience in effectively using the model cannot
be over emphasized. Experience gained through knowledge of the physical
systems being modeled and repeated applications of the model will
enhance the user's ability to choose the proper values of the various
parameters, e.g. timestep size. The user will also gain the ability to
anticipate the effect of changing the value of a particular parameter by
a certain percentage on the model solution (i.e. model sensitivity).
5.7. Model Applicability
5.7.1. Water Quality Problems
The model can be used to assist in the performance of the following
water quality related computational tasks:
1. ) Assessment of the disposition of dissolved and sorbed pollutants,
possibly either transported to an estuary or harbor by stormwater
runoff or released into these water bodies by nearby industries,
and their effect on the receiving waters and the aquatic ecosystem
therein, when linked with a particulate contaminant transport model
that contains a sorption submodel (Onishi and Wise, 1979).
2. ) Prediction of the effect of reduced sediment inflows to estuaries,
caused by upstream water storage and subsequent use, to ascertain
the degree of waste water management required to control estuarial
water pollution.
259
3.) Prediction of the limitation of sunlight penetration in estuarial
waters resulting from high turbidity levels which, in turn, are
caused by high concentrations of suspended sediment. This reduced
light penetration can cause the algae multiplication rate to
decrease significantly, and thus affect the entire aquatic
ecosystem.
5.7.2. Sedimentation Management Problems
The model can be used as a tool to help solve the following
sedimentation problems:
1. ) Prediction of the movement of dredged material released in open
waters in order to estimate the effect of the disposal at a given
location in the water body on the shoaling rates elsewhere, and in
particular in the dredged area.
2. ) Selection of harbor sites in estuaries and bays where shoaling is
minimized.
3. ) Prediction of changes in the sedimentary regime that may occur as a
result of a proposed change or development of an estuary or harbor,
such as the dredging of new navigation channels and the possible
change in the salinity field (e.g. further inland intrusion) caused
by the proposed change.
4. ) Estimation of shoaling rates and maintenance dredging requirements
in areas of very low flow such as marinas, harbors and docks, and
recommendation of means by which shoaling rates might be minimized.
5. ) Prediction of the spatial (primarily longitudinal) variance in the
shoaling and/or erosion rates, caused by varying flow conditions
and salinities, along the entire reach of an estuary.
CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
6.1. Summary and Conclusions
Cohesive sediments are comprised largely of terrigenous claysized
particles. The remainder may include fine silts, organic matter, waste
materials and small quantities of very fine sand. The electrochemical
surface repulsive forces which act on each elementary clay particle are
approximately six orders of magnitude larger than the gravitational
force. As a result, the physicochemical properties of cohesive
sediments are controlled mostly by these surface forces.
In water with a very low salinity (less than about 1 ppt) the
elementary particles are usually found in a dispersed state. A slight
increase in the salinity (up to 23 ppt) is sufficient to repress the
repulsive surface forces between the elementary particles, with the
result that the particles coagulate to form aggregates. Each aggregate
may contain thousands of elementary particles. Coagulation depends upon
interparticle collision and cohesion after collision. The three
principle mechanisms of interparticle collision in suspension are
Brownian motion, internal shearing and differential sedimentation.
Cohesion of elementary particles is caused by the presence of net
attractive surface forces. The latter condition is caused by the
increased concentration of dissolved ions, which serves to depress the
double layer around each particle and allow the attractive forces to
predominate.
260
261
Most estuaries contain abundant quantities of cohesive sediments
which usually occur in the coagulated form in various degrees of
aggregation. The transport of cohesive sediments in estuaries is
strongly influenced by the coagulation behavior of dispersed sediment
particles, which is controlled by the chemical composition of the
suspending fluid, the hydrodynamic conditions, the concentration of
suspended sediments and the physicochemical properties of the sediment.
Sediment related problems in estuaries include shoaling in
navigable waterways and water pollution. The mixing zone between upland
fresh water and sea water, as well as areas such as dredged cuts,
navigation channels, harbors and marinas are favorable sites for
sediment deposition. Since estuaries are often used as transportation
routes, it is necessary to accurately estimate the amount of dredging
required to maintain navigable depths in these water bodies. A
significant portion of the pollution load in a water body is typically
transported sorbed to cohesive sediments. Therefore, the importance of
considering the transport of these sediments in predicting the
disposition of pollutants introduced into an estuary cannot be over
emphasized.
A twodimensional, depthaveraged, finite element cohesive sediment
transport model, CSTMH, developed during this study may be used as a
tool in the field of estuarial management to predict the fate of sorbed
pollutants and the frequency and quantity of dredging required to
maintain navigable depths. CSTMH is a time varying model that is
capable of predicting the horizontal and temporal variations of the
depthaveraged suspended sediment concentrations and bed surface
elevations in an estuary, coastal waterway or river. The two
262
dimensional, depthaveraged advectiondispersion equation with
appropriate source/sink terms is solved at each timestep for the nodal
concentrations. Previous models are not as comprehensive as they use
mathematical descriptions (or algorithms) of the transport processes
(that are considered), that are based on limited studies conducted prior
to the early 1970 's. In this study field evidence and the considerable
amount of experimental research that has been conducted on the mechanics
of cohesive sediment transport since that time have been used to develop
new algorithms which describe the processes of erosion, dispersion,
settling, deposition, bed formation and consolidation. This has
resulted in a model whose oredictive capability is improved over that of
previous models. A summary of the algorithms and the improvements
achieved in the mathematical representations of these transport
processes is given next.
Deposited estuarial sediments occur in three different stages of
consolidation: unconsolidated, partially consolidated and settled
(fully consolidated). Unconsolidated deposits, referred to as
stationary suspensions, possess a very high water content and low shear
strength and are redispersed, or mass eroded, when subjected to an
excess bed shear stress. Partially consolidated deposits have a
somewhat lower water content and higher shear strength and are
resuspended aggregate by aggregate, i.e. undergo resuspension, when
subjected to an excess shear stress. Settled, or fully consolidated
beds possess a much lower water content, a much higher shear strength
and as well are resuspended aggregate by aggregate when subjected to an
excess shear. The shear strength and the density of partially
consolidated beds have been shown by laboratory tests to increase with
263
depth below the waterbed interface, and as such are vertically
stratified. Both stationary suspensions and partially consolidated beds
undergo consolidation due to overburden pressure, with the bed density
and shear strength increasing with time of consolidation. In settled
beds, the shear strength and the density profiles exhibit relatively
uniform properties over the depth. The sediment bed schematization
incorporated in CSTMH allows for the abovementioned three bed
sections, and divides each section into a characteristic number of
layers. Within each layer, the bed shear strength and density are
assumed to vary in a linear manner with depth. The number of layers as
well as the shear strength and density profiles in each section must be
determined from laboratory erosion tests. Even though a stationary
suspension is not a true bed, it is represented as such in order to
account for the sediment mass which forms this suspension. Previous
models use a constant bed shear strength and bed density for each layer,
and use only a single layer for the partially consolidated bed
section. Therefore, the stratified nature of partially consolidated
beds is not represented in these models.
The bed formation algorithm incorporated in CSTMH uses the assumed
linear bed density profile in each layer to iteratively solve for the
thickness of bed formed by the deposition of a given mass of sediment.
The bed structure (i.e. bed shear strength and density profiles) of the
existing bed is adjusted to account for the added sediment mass.
Previous models use the assumed constant bed density value in each layer
to solve explicitly for the bed thickness.
The erosion algorithm simulates the redispersion of stationary
suspensions by instantly redispersing the thickness of the bed above the
254
level at which the bed shear stress, Tj^, is equal to the bed shear
strength, x^. The average resuspension rate of the partially
consolidated bed layers over one timestep At is given by an empirical
law (Eq. 3.13) that is analogous to the rate expression which results
from a heuristic interpretation of the rate process theory of chemical
reactions. This rate expression indicates that the resuspension rate
varies exponentially with the excess bed shear stress. The average
resuspension rate of the settled bed layers is given by an empirically
determined expression (Eq. 3.12) that is equal to the first term of a
Taylor series expansion of the empirical resuspension rate law for
partially consolidated bed layers. Thus, the rate of erosion of settled
beds is linearly proportional to the excess shear. The thickness of the
partially consolidated bed section or the settled bed section eroded per
timestep is determined using an iteration routine. The effect of
salinity on the bed shear strength, and hence on the erosion rate of
that bed, as determined from laboratory resuspension tests is
incorporated into the erosion algorithm. For a natural mud, the bed
shear strength was found to double in value, in a linear manner, between
S = 0 and 2 ppt, and thereafter (for S > 2 ppt) was found to remain
practically constant. Based on an interpretation of typically observed
Eulerian concentrationtime records in estuaries, erosion is simulated
to occur only during temporally accelerating flows when t,^ is greater
than T^. Previous models: 1) assume that the erosion rate of both
partially consolidated and settled beds varies linearly with the excess
bed shear stress, 2) do not account for the effect of salinity on the
bed shear strength, 3) assume that erosion occurs whenever, in either a
temporally decelerating or accelerating flow, is greater than and
265
4) do not account for the bed shear strength and density profile 1n each
layer (since constant values for and p are used) in determining the
mass of sediment eroded.
The dispersion algorithm developed in this study utilizes the
Reynold's analogy between mass and momentum transfer and solves for the
four components of the twodimensional sediment dispersivity tensor
using the formulation derived by Fischer (1978) for bounded shear
flows. Thus, only shear flow dispersion is accounted for in this
algorithm. The limitations of such a dispersion algorithm are
consistent with those associated with a twodimensional, depthaveraged
cohesive sediment transport model. Previous cohesive sediment transport
models: 1) do not include the cross product dispersion coefficients in
the advectiondispersion equation and 2) do not include a dispersion
algorithm to calculate the dispersion coefficients as functions of the
local depth of flow and the depthaveraged velocity components.
The settling velocity of cohesive sediments is a function of, among
other parameters, the suspension concentration, C, the salinity, S, and
■z^. For concentrations less than C^^ = 0.1  0.7 g/1 the sediment
particles settle independently without much mutual interference, and
therefore the settling velocity is independent of C. In the range <
C < 10  15 g/1, the settling rate is proportional to C" with n >
0, due to mutual interference. In the range C > the settling
velocity decreases with increasing concentration due to hindered
settling.
The deposition algorithm integrates the concepts proposed by
various investigators and represents a unified model of this process.
Deposition is predicted to occur only in decelerating flows, i.e.
266
tjj(t+At) < "^bCt), when is less than the maximum shear stress at which
deposition can occur, \ . For \ < \ r Change I), where  is the
''max u u,^
value of '^^ at which the deposition rate in Range I is equal to that in
Range IIB (defined below), and for C < Ci for all values of \ < \ ,
"max
the rate of deposition is determined using the exponential law given by
Eq. 3.28. For \ r ^ \ ^ \ ^nd C > C, (Range IIB), the deposition
"»' "max ^
rate is given by a lognormal expression (Eq. 3.45). The thickness of
the bed formed by a given deposited sediment mass is determined using
the properties of the unconsolidated and partially consolidated bed
sections. As deposition continues, first the unconsolidated layers are
filled up, followed by the partially consolidated layers.
Increasing the salinity of the suspending fluid was found to
increase slightly the settling velocities, and hence the deposition
rates of a natural mud. The settling velocity, W^, was found to
increase as s'^*^^ in Range I. The effect of salinity on the deposition
rate in Range IIB is given by Eqs. 3.52 and 3.54, and was approximately
the same as that in Ranges I and IIA.
The settling/deposition algorithm in previous models: 1) do not
include the effect of salinity on the rates of deposition and 2) predict
that deposition occurs only when t^k ^ "^h » either temporally
min
accelerating or decelerating flows. Thus, deposition is predicted to
occur in the previous models during only a small percentage (e.g. 20%
for kaolinite in tap water) of the shear stress range in which
deposition has been observed to occur in laboratory steady flow
experiments.
The consolidation algorithm accounts for the consolidation of a
stationary suspension and partially consolidated bed by increasing the
267
bed density and bed shear strength with time. Consolidation is
considered to begin after the bed formation process is complete, at
which time the bed thickness will be maximum. After two hours of
consolidation the stationary suspension layer(s) become part of the
partially consolidated bed, and therefore would undergo resuspension if
subjected to an excess shear stress. The variation of the mean bed
density with consolidation time is given by Eq. 3.67, while the density
profile p{zb) is determined using power law relationships between p and
(Eqs. 3.703.72). The thickness of the bed is reduced to account for
the expulsion of pore water during consolidation, and to insure that the
mass of sediment in the bed is conserved.
The new deposit bed section of the layered bed model is further
divided into a finite number of strata in order to account for repeated
periods of deposition, as typically occur in estuaries due to the
oscillating tidal flow. The top stratum may be composed of a stationary
suspension and partially consolidated bed, whereas the buried strata are
composed of just partially consolidated sections. The degree of
consolidation of a particular stratum is accounted for by using a
separate consolidation time for each stratum.
Due to the extremely limited information on bed shear strength
profiles in cohesive sediment beds, the variation of ^(zk) with
consolidation time is determine using a power relationship between p and
c^, as given by Eq. 3.61.
None of the previous cohesive sediment transport models account for
the increase in the bed shear strength and density profiles due to
consolidation, and thus are not capable of simulating the decrease in
the susceptibility to erosion of a consolidating bed with time due to
the continual increase in the bed shear strength.
268
The Galerkin weighted residual method is used to solve the
advectiondispersion equation for the nodal suspended sediment
concentrations. An existing finite element formulation was modified to
include the two cross product dispersion coefficients. and D . The
Aj y X
model yields stable and converging solutions. The accuracy of the
solution is affected when the Peclet number becomes too large (greater
than lo2) or too small (less than 10"^).
Verification of CSTMH was carried out against four erosion
deposition experiments, three of which were performed in an 18.1 m long
recirculating flume and the fourth in an 0.76 m mean radius rotating
annular flume. Simulation of a deposition experiment in an 100 m long
nonrecirculating flume at the Waterways Experiment Station, Vicksburg,
Mississippi yielded satisfactory agreement between the measured and
predicted deposit thickness in the downstream 65 m reach of the flume.
The recorded differences in the upstream 35 m reach may have been caused
by the occurrence of longitudinal sorting, or the increased turbulence
present in this section of the flume. CSTMH is not capable of
simulating longitudinal sorting because the parameters which
characterize the lognormal depositional law, used for deposition Range
IIB, are assumed to be spatially invariant.
The model was applied to prototype conditions. Sedimentation in
Camachee Cove Marina, located adjacent to the Intercoastal Waterway in
St. Augustine. Florida was modeled. A mean shoaling rate of 8.6
cm/year, which is representative of fair weather conditions only, was
predicted. Sediment transport in a 10 km hypothetical canal, in which
both erosion and deposition occurred, was also simulated at four
different salinities to show the effect of salinity on the rate of
269
sediment transport under typical prototype conditions. These three
simulations demonstrated the significant influence of salinity on the
rate of erosion of a cohesive sediment bed for salinities less than 10
ppt, and the reduced effect for salinities greater than about 10 ppt.
6.2. Recommendations for Future Research
Based on the conclusions from the present study, the following
objectives and recommendations for further research are made:
1. Conduct a thorough laboratory investigation of the
consolidation characteristics of cohesive soils. Objectives for such a
research program could be the following: 1) verify and/or modify the
multi strata bed formationconsolidation algorithm developed during this
study. 2) Determine the validity and practicality of using one of the
existing explicit finite difference finite strain consolidation models
to predict the consolidation of cohesive soils. 3) Investigate further
the possible correlation between the density and shear strength of
cohesive soils.
2. A laboratory investigation of the redispersion characteristics
of stationary suspensions and partially consolidated beds is
recommended. This recommendation is motivated by the differences
obtained between the measured and predicted suspension concentrations
during the first onehalf hour of periods of erosion in the three
experiments conducted in the 18.1 m recirculating flume. Erosion
experiments using flow deposited cohesive sediment beds should be
conducted in which the consolidation time and applied bed shear stress
are systematically varied.
270
3. The results from the model simulation of the tidal cycle
experiment in the rotating annular flume reveal the need for studying
the variation of the vertical suspended sediment profile in an unsteady
flow field. It would be advantageous to conduct such an experiment in a
rotating annular flume because longitudinal velocity and suspended
sediment gradients do not occur in this facility.
4. The deposition algorithm developed during the present study can
be modified to include the effect of longitudinal sorting on the rates
of deposition by incorporating the variation of tgg and cr^ in the two
horizontal dimensions. Extensive field and laboratory tests would be
required to determine the following relationships: t^Q = tgg (x.y.x^)
and 0^ = {x,y,x^).
APPENDIX A
DERIVATION OF ADVECTIONDISPERSION EQUATION
In a diffusing mixture such as the sedimentwater binary system,
the various constituents move at different velocities. For example, the
vertical advective velocity of the water differs from that of the
negatively buoyant sediment particles by the sediment setting velocity,
Wg. The local mass averaged hydrodynamic velocity for a binary system
is defined as that which would be measured by a pi tot tube:
^ total momentum C V.. + C.V,
(A.l)
Y  WW s s
mass of mixture C + C
w s
Where = velocity of water mass, V = velocity of sediment mass, C
3 ' w
mass of water/total volume of mixture and = mass of sediment/total
volume of mixture. Using the coordinate system defined in Fig. 3.1, Y
and are defined as
\ = ui + vj + wk (A. 2)
= ui + vj + {w+W^)k (A. 3)
where u, v and w are the fluid velocity components in the respective
Cartesian coordinate directions. Here the water and sediment particles
are assumed to be advected in the x and y directions at the same
respective velocity components.
271
272
The advective mass flux of sediment is defined as
 %\ (A.4)
while the diffusive mass flux is given by
J„ = C (V V)
s  S^'s"^' (A. 5)
The diffusive flux in Eq. A. 5 is that due to molecular diffusion,
which by Pick's first law is equal to
's V() (A.6)
P
where p = density of binary system and d^^ = molecular diffusivity of
the sediment particles in water. In general D^^^ is a function of Cj,
C^, ^, T and p, with the latter three parameters being the absolute
viscosity, temperature and pressure of the binary system. The total
flux of sediment, N3, is equal to the sum of the advective and diffusive
fluxes, given by Eqs. A.4 and A.6, respectively.
h ~ ^s = ^s^  PDsw'() (A.7)
P
The law of conservation of sediment leads to the following
continuity equation for suspended sediment:
ac
^+^.H =S (A.8)
at ^
273
where S = source/sink term to account for the mass of sediment addded or
removed per unit volume per unit time. Substituting Eq. A. 7 into Eq.
A. 8 gives
ac * C
~^''^s\= ^•P'^sw'^^ ^ S (A.9)
ot p
In tensor notion Eq. A9 becomes
ac 9 a 5 c
at ax. ^ ^ dx. ^^ax. p
' J
In turbulent flow the instantaneous velocity components and
suspended sediment concentration can be expressed as the
sum of a timeaveraged term and a fluctuating component, i.e.
V, = + v'
s. s^ s.
" ^s S (A. 11)
where the overbar and the superscript prime denote the mean and
fluctuating quantities, respectively. The mean term is averaged over a
time interval, Tj, which is small compared with the time scale for the
mean flow, but large compared with the time scale for the turbulent
fluctuations so that the time averages of v' and c' over Tt are
^ J 5 1
approximately zero (MacArthur, 1979). Substituting Eq. A. 11 into Eq.
A. 10 and averaging the entire equation over Tt gives
ac a   a , . a d c
— + (C V ) + (C Y ) = [pD ()] + S (A. 12)
St ax. ^ ax 5x^ swg _ p
274
where the subscript s on the sediment concentration C3 has been dropped
for convenience, and where the following definitions have been used:
 1 '''l 1 ^^"^I .
J C dt c = — / C dt = 0
(A. 13)
Tj t Tj t
1 t+T T t +T
V = — / V dt V = — / V dt = 0
i Tj t ^ Tj t ^
The terms C V^. in Eq. A. 12 represent the turbulent diffusive mass
transport of sediment due to the turbulent velocity fluctuations in the
x^. direction. Reynolds analogy which is based upon the analogy between
the transfer of mass and momentum in turbulent flow and upon
Boussinesq's eddy viscosity hypothesis is used to relate these diffusive
sediment transport terms to the spatial gradient of the timeaveraged
concentration as follows:
I . 5C
C V = E /, ,.v
s. X. ^ (A. 14
1 1 Qx.
where E^. = turbulent diffusion coefficients of sediment in the x
direction. Substituting Eq. A. 14 into Eq. A. 12 gives
ac a a a c a 9r
Z^Z'^'^^'^i^ ^r^P'^swr^)^ ^— (E, — ) + S (A. 15)
°t ex. 1 dx ax. p ax. ^i Sx.
where the bars over the mean parameters have been dropped for
convenience.
275
Next, the fraction of sediment mass, C/p, 1n the molecular
diffusion term in Eq. A. 15 is simplified by effectively assuming that
the mixture density or total mass concentration is not a function of
position in the binary system. The physical justification for this
assumption is that the effect of spatial gradients of p is included in
the molecular diffusivity, D^^. As a result, Eq. A. 15 may be rewritten
as
9C a 8 5C
— + — (CV ) = (E ) + S (A. 16)
at ax. ^• dx. ^ax^. ,
where E^j = D^^ + E^^^ is the turbulent diffusion tensor, in which the
offdiagonal terms of . are not neglected.
Equation A. 16 is the threedimensional form of the advection
dispersion equation for suspended sediment transport in a fully
developed turbulent flow field. Upon expansion of the tensor terms. Eq.
A. 16 becomes
ac a a 5 a ac
— + — (Cu) + — (Cv) + — {C(w+W )} = — (E — +
at ax dy 92 ^ ax ^^ax
ac ac a 5c ac sc
a ac ac ac
— (E_ — + E^ — + E — ) + S
az ^^ax ^^ay ^^az
The desired twodimensional form of Eq. A. 17 is obtained by
integrating this equation over the local flow depth d, which is defined
as
276
d = h(x,y,t)  b(x,y,t)
(A. 18)
where h(x,y,t) and b(x,y,t) = elevations of the water surface and bed,
respectively, with respect to a tidal datum. When vertical integration
from b(x,y,t) to h(x,y,t) is performed, vertical profiles of the time
averaged velocity components in the x and y directions and the suspended
sediment concentration give rise to dispersion terms. In order to
account for such terms, the velocity components, u, v and w, and the
concentration, C, are expressed as the sum of a depthaveraged term and
a term which is the deviation of the parameter over the depth of flow
about the depthaveraged value, e.g.
~ II
V = V + V
(A. 19)
C = C + c"
where the double bar and the double prime denote the depthaveraged and
the deviating quantities, respectively. The following definitions of
the depthaveraged and deviating terms are used:
e =  J e dz with / e dz = o (a. 20)
d b b
for 9 = u,v,w, and C.
Equation A. 19 is substituted into Eq. A. 17 and the entire equation
is integrated from b{x,y,t) to h(x,y,t) using Leibnitz rule. The result
is given on a termbyterm basis below:
u + u
w = w + w
277
h ac h a = „
/ _dz = / — (C+c )dz
b at b at
ab
c"(b,t)—
at
a = = a(hb)
— {C(hb)}  c
at at
ah
c (h,t) — +
at
(A. 21)
ha h a = „ = „ a
J — (Cu)dz = / — {(C+C ){u+u )}dz = — {Cu(hb)} 
b ax b ax 5x
..= ah = 5h ., 5h
{Cu} _ {c ull —  {cu } —  {c u } — +
h ax h ax h ax h ax
a h „ „
— / C u dz
ax b
(A. 22)
ha a == == 5h ,.= dh
J — (Cv)dz = — {Cv(hb)}  {Cv}l {C v}
b ay dy h ay h ay
ah „ „ 5h a h „ „
{Cv } {c V } — + — / c V dz (A. 23)
h ay h ay dy b
ha =.. = ..
J — (Cw)dz = {(c+c )(wH^ )} (A 24)
b az h
h a
/ — (CW Jdz = {CW }  {CW }!
b az 5 2 h ^ b
ha ac a ac ac a
b ax ^^dx dx ^^ax ^^Bx ax
>■ II
ac ah ac ab a „ ah
{E ■ } _+{E } {E [c] — 
''''ax h ax ''''ax ^ ax ax ^ ax
„ ab
t^c \ — )} (A. 25)
ax
278
J — (Evv— = — {(hb)E, — }  (E, — )— (hb) 
b ay y^dy
ay
yy
ay
yy
ay ay
ac ah ac ab a „ ah
y^ay h ay yyay b ay ay yy ^ Sy
„ ab
ay
(A. 26)
h a ac ac 5c
/ — )dz = {E^^ }  {E }
b az az ^ az b
(A. 27)
h a dc a ac ac a
/ {E — )dz =_{(hb)E — }  (E — )— (hb) 
b ax ^yay ax ^yay
ay ax
ac
ah
ac ab a
ah
ay h ay
xy
ay b ax ax
xy
'ay
ab
ay
(A. 28)
h a ac a ac ac a
J ^(Eyy— = — {(hb)E^— }  (E — )— (hb)
b ay ^yax
ay
xy
ax
xy
ax ay
ac ah ac ab „ ah
^yax h ay ^yax bay ^y "ax
„ Bb
Cc 1— )}
''ax
(A. 29)
ha ac a h ac
/ — (E _)dz = —
V ax ^^az
/ E
ax b ^^az
ac
 {E } +
XZa u
az h
279
9c
(E }l
OZ b
(A. 30)
ha ac a h ac
/ — (E^ ~)dz = — / E —
b Sy y^az dy b y^az
II
ac
y^az b
ac
 {E — } +
y^az h
(A. 31)
h a 5c a = „ 5 =
/ — )dz = — (C+c )]  {E — (C+c")}i
h az '^^ax ^^ax h ^^ax b
(A. 32)
ha dc
5 =
5 =
ay
(A. 33)
Substituting Eqs. A. 21 through A. 33 into Eq. A. 17 and dropping the
double overbars from the depthaveraged terms gives
a a a ahnu ah
— (dC) + — (dCu) + — (dCv) + — / u"c" dz + — J v"c"dz=
dy ax b dy b
a ac ac a 6c ac
■^^^yy + } + {dE — + dE — } +
ax ^^ax >^yay ay ^^ax y^ay
" "
a h ac a h ac
— / dz + — / E dz +
ax b ^^az 8y b y^ay
h
H + B + / Sdz (A. 34)
b
280
where H represents the sum of the outward normal flux of sediment and
change in storage at h(x,y,t), and B represents the sum of the normal
sediment flux out and storage change at b(x,y,t). The expressions for H
and B are given below:
[C+c ] — + [(c+c ){u+u )!—+ [(C+c )(v+v )]^—
"at %x ■'dy
II II
ah
[(c+c )(w+w )\  C(C+c )W 1.  E [—(C+c )]^
s n XX
ax
'ax
ah
ac
ah
Eyy[— (C+c )],— + [E_ ],  E_[(C+c )],— 
ay
ay ^^sz ^ y^'
'ax
ah
ac
ac
E^y[— (C+c )\~  [E.,_ ]
ax
■ay
^ 11 a „ a „ ah
[E — (C+c )] + [E — (C+c )]  — {E„Jc +
X^5x " y^ay
ax
I, ah
,1 ah
ay
,1 ah
ax'
(A. 35)
II ab „ a „ 5b
B = [C^C [(Cc )W^], . E^/(C*c )]^.
ax ax
'at
a II ab ac a „ ab
EyyMc+c \— [E — \ + E [— (C+c )].— +
yy ?^^. D^.. ^ y^ ay l^ax
ay °ay
a „ ab ac ac
ax
^ay " ^"^^^az ' ^^y^az'b
° .1 a 5 „ ab
[E.,— (C+c )]  [E —(C+c )] + — {E [c ]— +
xz
ax
ay
^ ax ^x
281
I. 9 „ Sb 9b
The terms H and B represent the boundary conditions at h(x,y,t) and
b{x,y,t), respectively, since Eq. A. 17 was vertically integrated from h
to b. H is equal to zero since it is assumed that there is no net rate
of transport of sediment across the instantaneous free water surface.
Therefore, Eq. A. 34 can be simplified to yield
0 0 9 11 ■ II 9
— (dC) + u — (dc) + V — (dC) + — (du"c") + — {dv"c") =
at 9x 9y dx 9y
a 9c 9c 9 ac ac
~~^'^^yx~ ^^vv— ^ + — ^dE ~ + dE — } +
ax ^^9x ^y9y 9y y>^ax Qy
» II
a h 9c 9 h ac h
Z I ^z7^' r / ^z7^^ ^ ^ ^ Sdz (A. 37)
9x b 9z 9y b b
where the double overbar denotes the depthaveraged value of the
quantity thereunder, and where the assumption of an incompressible fluid
(i.e. V.u = 0) has been utilized.
A Reynolds analogy is again used to relate the quantities under the
double overbars in Eq. A. 37 to the spatial gradient of the depth
averaged suspension concentration as follows:
II II 9c
u.c = K. . —
1 IJ;
'ax,
1 .J
1,2
(A. 38)
282
where K^j = twodimensional sediment dispersion tensor. Substituting
Eq. A. 38 into Eq. A. 37 gives
a 5 a a 5c 5c
— (dC) + u— (dC) + V— (dC) = — {dD — + dD — } +
at Qx 5y dx ^^dx ^^dy
a ac ac a h ac
— {dD — + dD — } + — / E^^ dz +
ay y^ax y^ay dx b ^^^z
II
a h ac h
— / E dz + B + / Sdz (A. 39)
ay b y^az b
where D^j = K^.j + Z. ■ = effective sediment dispersion tensor.
If Ey2 and Ey^ are both assumed to be linear functions of z such
that the partial derivatives of E^^ and Ey^ with respect to z are
functions of only x and y, respectively, the two integrals in Eq. A. 39
become
a h ac a
ax b ^^^z ax X2 h xz b
(A. 40)
s h ac a
(A. 41)
The first and second terms on the right hand sides of Eqs. A. 40 and
A. 41 should be incorporated into H and B, respectively, as they
represent fluxes of sediment out of the water surf act and bottom.
283
Therefore, Eq. A. 39 becomes
9 5 a 5 5C dC
— (dC) + u— (dC) + V— (dC) = — {dD — + dD — } +
at 9x 5y ax ^^ax ^^ay
B ac ac
where S = depthaveraged source/sink term.
The term B in Eq. A. 42 represents the rate of erosion and
deposition at the bedfluid interface, b(x,y,t). That is, sediment that
settles or diffuses out of suspension is part of the depositional flux,
and sediment that advects or diffuses into suspension is part of the
erosional flux. Therefore, B can be alternatively expressed as
d d
B = e + Ti = — (dC) +— (dC) (A. 43)
dt Erosive dt Depositional
Flux Flux
in which e = idC)] _ = dry mass of sediment eroded per unit
dt Erosive Flux
bed surface area, and ti = MdC) ^ ^^^^ ^ ^^^^^ sediment
dt
deposited per unit time per unit bed surface area.
The depthaveraged source/sink term S represents the rate of
sediment influx to or outflux from the water body not attributable to
either the processes of erosion or deposition. For example, S would
account for the removal (sink) of a certain mass of sediment by dredging
in one area (e.g. navigational channel) of a water body, and the dumping
(source) of the sediment as dredge spoil in another location in the same
body of water.
284
So Eq. A. 42 is the twodimensional, depthaveraged advection
dispersion equation which governs the transport, addition (i.e. source)
and removal (i.e. sink) of suspended sediment in a turbulent flow
field. Expressions for £ and ti are given in Sections 3.4 and 3.6,
respectively.
The total dispersive fluxes, f, in the x and y directions are
given by
ac ac
f„ = dD — + dD — (A. 44)
X xxg^ xy^y ^'^•'^^^
ac ac
f = dD — + dD — (A 4"^^
y yxa^ yy^^ ^'^•^s)
which are the expressions inside the curly brackets in Eq. A. 42. Using
the coordinate system shown in Fig. 3.1, the dispersive fluxes in the
longitudinal (along the flow axis) and transverse (perpendicular to flow
axis) directions, f, and f^, respectively, are related to f, and f„ as
' X y
follows:
f^cos Q  f^sin e
fy = f^sin 9 + f^cos e
(A. 46)
(A. 47)
where f, and f^ are given by
ac
f, = dD, — (A. 48)
ac
f^ = dD — (A. 49)
an
285
in which © = arctan (u/v), Dj = effective longitudinal dispersion
coefficient and = effective transverse dispersion coefficient. The
four components of the twodimensional dispersion tensor, D^j, are
related to D] and by the following functional relationships:
D^^ = D^cos^e + D^sin^e (A. 50)
'^xy = V =V^D^D^)sin(2e) (A. 51)
D = D^sin^e + D^cos^e (A. 52)
Expressions for D^j are given in Chapter III, Section 3.5.
APPENDIX B
COEFFICIENT MATRICES IN THE ELEMENT MATRIX
DIFFERENTIAL EQUATION
The purpose of this appendix is to list the matrices and vectors in
the element matrix differential equation (4.23) and describe how they
and the contour integral in Eq. 4.22 are evaluated.
The product of the element coefficient matrix, [k], and the nodal
concentration vector, {C}^, is seen from Eqs. 4.22 and 4.23 to be equal
to
A
.dc .5c ^'U ac
[k]{c}^ = // [N (u— + v— ) + [D — +
e ax ay 5x ^^ax
ac ac^ dc
where the approximate velocities, u and v, are evaluated at each point
(C,Ti) as follows:
l=n ^ i=n
u = 2 N.u. v = 2 N.v. (B.2)
1=1 ^ ^ 1=1 ^ ^
The dispersion coefficients are considered not to vary significantly in
either space or time in this formulation, and therefore are assumed to
be constants. The transformation from global coordinates to local
element coordinates derived in Chapter IV, Section 4.4.2, gives
286
287
dxdy = lJ5dTi
{B.3)
Substituting Eqs. B.2 and B.3 into Eq. B.l and dividing through by the
concentration vector {C}^ gives the following expression for the (i,j)
term of [k]:
' ^ ^ ax ^ ^ay ax ax
aN. an . aw. aN . an. aN.
a7 ~ ^y'^ a7 aJT "^yy a7
JdCdTi for 1=1, n (B.4)
The notation 1=1, n indicates that the variables with the subscript 1 are
summed from 1=1 to l=n=number of nodes in a particular element. Thus,
the element coefficient matrix is given by
i=n j=n l=n
[k] = 2 Z E k (B.5)
i=l j=l 1=1
The double integration in Eq. B.4 is performed using the Gauss
Legendre quadrature formula as follows (Ariathurai et a1_., 1977):
. , m=NQ mm=NQ
/I i_i f(^,il) d^dTi = S 2 H H f( ,Ti ) (B.6)
^ ^ m=l mm=l ™
288
where f=k Is given by Eq. B.4, NQ= number of quadrature points and
H=weight factors.
The element temporal matrix [t]^ is given by
i=n j=n l=n
Itl = >: z z // N H N d, [J! d^dn (b.7)
i=l j=l 1=1 % 1 J ' I
where again the integration is performed using the GaussLegendre
method.
e
The element source/sink vector {f} is given by
i=n j=n l=n
if} =  ^ Z Z // N.NTd,SJdldn (B.8)
i=l j=l 1=1 e ^ ' '
and the boundary matrix [b]^ is given by
i=n j=n l=n ' j
2 2 E / M N d, [(D + D^^ ) n^
i=l j=l 1=1 ^ 1 ' XX 5x ay X
The boundary matrix [b]e accounts for a specified concentration flux
boundary condition along the boundary of a domain boundary element.
APPENDIX C
COMPUTER PROGRAM
The computer program of CSTMH is written in FORTRAN IV using
double precision arithmetic. Double precision is required in simulating
laboratory scale tests in order to minimize roundoff error. Prototype
systems can be modeled using single precision arithmetic. A description
of the main program and the subroutines followed by a flow chart and
user's manual are presented.
C.l. Main Program
In the main program the following information is read:
input/output file numbers, problem option and control parameters, finite
element grid geometry and transient control parameters. The I/O file
numbers determine the file numbers used in both reading and printing
certain data. The problem option parameter, NOPT, specifies which one
of four types of problem is to be solved. The four types of problems
that can be solved by CSTMH are: 1) steady state advectiondispersion
of a conservative constituent, 2) unsteady advectiondispersion of a
conservative constituent, 3) cohesive sediment transport and 4) one
dimensional consolidation of a clayey sediment bed. The flow chart in
Section C.3 diagrams the program steps used in the solution of each
problem type. The transient input code arrays specify at which time
steps new boundary conditions, flow depths, velocity field, dispersion
289
290
coefficients, sediment settling velocities, density (i.e. salinity)
field or local source/sink terms (explained in Appendix A) are read in
or calculated using either builtin algorithms or user supplied
routines. The user's manual in Section C.4 gives a short definition of
each parameter read in either the main program or one of the
subroutines. The grid data that need to be read includes the number of
nodes, NP, number of elements, NE, nodes forming each element (nodal
connections in counterclockwise direction) and the two horizontal
coordinates of each node.
The main program also initializes necessary arrays, prints out the
initial conditions and initial values of certain parameters, contains
the main time loop and prints out the concentrationtime record for
specified elements.
C.2. Subroutines
A brief description of the subroutines and subf unctions in CSTMH
is given below.
Subroutine BAND  Forms and solves the system matrix equation at each
time step for the nodal concentrations using Gaussian elimination. This
subroutine is used when ISOLV = 0.
Subroutine BEDFOR  Forms the sediment bed that is a result of 1)
deposition during the previous time step or 2) new deposits present on
top of the settled bed at the start of the modeling.
Subroutine BEDMOD  Control program for bed formation and consolidation
routines. Determines mass of stationary suspension for which T^^^, = 2
hours; calls Subroutine BEDFOR, and stationary suspension becomes part
of the partially consolidated bed. Determines the consolidation time
291
for each bed stratum, and calls Subroutine CONSOL for the strata with
Tjj, > 2 hours.
Subroutine BEDSS  Computes the bed shear stress at each node.
Calculates the average flow depth, velocity, bed shear stress, water
density and concentration for each element using the element shape
functions. Compares this time step's bed shear with the previous time
step's bed shear to determine if the flow at each node is temporally
steady, accelerating or decelerating.
Subroutine COMPAR  Compares analytical solution with numerical solution
for steady state problems.
Subroutine CONCBC  Reads or computes concentration boundary conditions
at specified nodes.
Subroutine CONCIC  Reads or computes the initial suspended sediment
(for sediment transport problem) concentration at every node.
Subroutine CONSOL  Computes the increase in the bed density profile in
each stratum due to consolidation. The bed strength profile is computed
as a function of the new bed density profile.
Subroutine DENSITY  Computes the water density at every node using the
given water temperature, salinity and suspension concentration. The
kinematic viscosity is also calculated as a function of the water
temperature.
Subroutine DEPMAS  Computes the dry sediment mass deposited during the
previous time step for every element where deposition is predicted to
occur.
Subroutine DEPSN  Computes the rate of deposition at each node where
deposition is predicted to occur.
292
Subroutine DEPTH  Reads or computes (using user specified procedure)
the depth of flow for each node at the time steps where the appropriate
transient code array indicates that a change in depth occurs.
Subroutine DISPER  Reads or computes (using the dispersion algorithm)
the four components of the twodimensional dispersion tensor for each
node at every time step where the appropriate transient code array
indicates that the values of the dispersion coefficients change.
Subroutine DRYNOD  Determines at which nodes and elements the water
depth is negative (i.e. dry). These nodes are eliminated from the
system array coefficient matrices.
Subroutine ELSTIF  Forms the element coefficient and load matrices.
Modifies element load matrix to account for specified boundary
conditions.
Subroutine EXACT  Computes analytical solution to steady state
convectiondiffusion problem.
Subroutine FRONT  Forms and solves the system matrix equation at each
time step for the nodal concentrations using the frontal elimination
routine. This subroutine is used when ISOLV = 1.
Subroutine ITERC  Computes thickness of partially consolidated bed
formed by specified dry mass of sediment using an iteration procedure.
Called by Subroutine BEDFOR.
Subroutine ITERM  Computes thickness of stationary suspension formed by
specified dry mass of sediment using an iteration procedure. Called by
Subroutine BEDFOR.
Subroutine LOAD  Forms the array NBC which numbers the equations in the
system matrix. Number of equations is equal to the number of nodes
minus the number of nodes at which boundary conditions are specified.
293
Computes the band width for the system coefficient matrix. This
subroutine is called when Subroutine BAND is used to solve the system
matrix.
Subroutine LOADX  Forms the array NBC. This subroutine is called when
Subroutine FRONT is used to solve the system matrix.
Subroutine ORGBED  Reads the original settled bed profile and the
initial dry mass per unit bed surface area of new deposits on top of the
settled bed for elements where such exists.
Subroutine RECORD  Records the values at each time step of various
parameters for the elements where time records are desired.
Subroutine RED  Called by Subroutine FRONT to read data from temporary
disc data file.
Subroutine REDISP  Computes the redispersion rate for unconsolidated
new deposit layers (stationary suspension) when the flow is accelerating
and the bed shear stress is greater than the shear strength of the
suspension surface. Computations are made on an elementbyelement
basi s.
Subroutine RESUSP  Computes the resuspension rate for exposed partially
consolidated bed layers or settled bed layers when the flow is
accelerating and the bed shear stress is greater than the shear strength
of the bed surface. Computations are made on an elementbyelement
basi s.
Subroutine SEDPRP  Reads the settling velocity, new deposit and
consolidation properties of the cohesive sediment. Settling velocity
parameters and the new deposit properties are printed out.
Subroutine SETVEL  Reads or computes using a builtin algorithm the
sediment settling velocity in Range I and for C < Ci in Range II as a
294
function of the suspension concentration and salinity at each node for
the time steps where the appropriate transient code array indicates that
a change in the settling velocity occurs.
Subroutine SHPFNS  Computes the isoparametric quadratic shape functions
and their derivatives for quadrilateral elements with parabolic sides.
Subroutine TSHAPE  Computes the isoparametric quadratic shape functions
and derivatives for triangular elements with parabolic sides.
Subroutine VEL  Reads or computes using a user specified routine the
depthaveraged components of the velocity in the x and y directions at
each node for the time steps where the appropriate transient code array
indicates that a change in the velocity occurs.
Subroutine WRITER  Prints out the bed shear stress, bed elevation,
erosion/deposition rates and the layerbylayer bed properties for each
element at each time step where the appropriate transient code array
specifies.
Subroutine WRT  Called by Subroutine FRONT to perform mass transfer of
data to temporary disc data file.
Function DENFUN  Computes the water density as a function of
temperature and salinity.
Function FBETA  Computes the value of the empirical coefficient A in
Eq. 3.68 as a function of T^^..
Function FDELTA  Computes the value of the empirical coefficient B in
Eq. 3.68 as a function of J^^.
Function FMBDN  Computes the value of the final mean bed density (given
by eq. 3.64) as a function of Cq.
Function FTCIN  Computes the value of T^^^.^^ (given by Eq. 3.65) as a
function of Cq.
295
Function FZHMIN  computes the value of z^^^^ (given by 3.69 with 9 =
z^^^) as a function of T^^,.
Function SIGFUN  Computes the value of 02 (given in Eq. 3.36) as a
function of t;^.
Function T50FUN  Computes the value of tgg (given in Eq. 3.36) as a
function of Tj^.
C.3. Flow Chart
296
START
J
READ I/O
FILE NUMBERS
COORDINATES
NO
297
0
1
/
READ TRANSIENT
INPUT DATA
READ ELEMENT
NUMBERS FOR WHICH
TIME HISTORY IS TO
BE PRINTED OUT
INITIALIZE
NECESSARY
ARRAYS
READ AVERAGE
WATER TEMP.
AND INITIAL
SALINITIES
READ SOURCE/SINK
TERfi AT
APPROPRIATE NODES
YES
SEDIMENT PROBLEM 
INITIALIZE BED
PROPERTIES
5
6
299
i
CALL DENSTY
ELEMENTAL BED SHEAR
STRENGTHS AND BULK
DENSITIES CALCULATED
AS A FUNCTION OF '
ELEMENTAL SALINITY
VALUES
300
CALL VEL
READ INITIAL
VALUES OF FLOW
VELOCITIES
CALL BEDSS
CALCULATE
NODAL
BED SHEAR
STRESSES
CALL DISPER
READ/CALCULATE
INITIAL DISPERSION
COEFFICIENTS
CALL SETVEL
YES
READ/CALCULATE
NODAL SETTLING
VELOCITIES
301
1
CALL
LOAD
NUMBER OF EQUATIONS
IN THE SYSTEM COEF.
MATRIX AND
BANDWIDTH
DETERMINED
CALL DRYNOD
DETERMINES WHICH
NODES AND
ELEMENTS ARE DRY
CALL LOADX
NUMBER OF EQUATIONS
IN THE SYSTEM COEF.
MATRIX AND BANDWIDTH
DETERMINED
PRINT INITIAL
CONDITIONS AND
SEDIMENT PROPERTIES
STEADYSTATE
YES
PROBLEM
NO ^
UNSTEADY AND
SEDIMENT
PROBLEfIS
YES
IF
ISOLV
EQ
NO
NOPT
EQ
NO
YES
302
CALL FRONT
NODE SOURCE
TERMS ADDED,
GLOBAL COEF.
MATRIX FORMED
AND SOLVED
BY FRONTAL
ELIMINATION
ROUTINE USING
FULL PIVOTING
YES
CALL BAND
NODE SOURCE TERMS
ARE ADDED INTO
SYSTEM LOAD
MATRIX, GLOBAL
COEF. MATRIX FORMED
AND IS SOLVED
BY GAUSSIAN
ELIMINATION
CALL DEPSN
DEPOSITION
RATES
CALCULATED
CALL RESUSP
SURFACE
EROSION
RATES
CALCULATED
NO
CALL
WRITER 
PRINT
CONCS.
CALL ELSTIF
ELEMENT STIFFNESS
ARRAYS FOR FIRST
TIME STEP FORMED
CALL COMPAR
COMPARE WITH
EXACT SOL.
MAIN TLME LOOP
DO N=2,iNTTS
DEPENDING ON INPUT
CODES, READ NEW
PARAMETERS FOR
THIS TIME STEP
CALL DENSTY
SET NEW
SALINITIES
CALL CONCBC
SET NEW
BOUNDARY
CONDITIONS
CALL DEPTH 
SET NEW
FLOW DEPTHS
YES
CALL DISPER
SE! NEW
DISPERSION
COEFFICIENTS
CALL SETVEL
SET NEW
SETTLING
VELOCITIES
READ NEW
NODAL SOURCE/
SINK TERMS
305
" CALL
DRYNOD
CALL 1
OADX
YES
CALL DEPSN
CALL REDISP
REDISPERSION
RATES CALCULATED
CALL RESUSP
CALL BAND
CALL FRONT
306
IF
iNOPT
EQ
NO
YES
CALL DEPSN
DEPOSITION
RATES CALCULATED
CALL BEDMOD
CONTROLS BED
CONSOLIDATION
ALGORITHM
CALL RECORD
OUTPUT FOR THIS
TIME STEP SAVED
II
307
308
C.4. User's Manual
SET A
CARD A.l
15
610
1115
1520
2125
2630
3135
3640
4145
4650
5155
5660
6165
CARD A. 2
1
278
CARD A. 3
15
JOB CONTROL CARDS
610
1115
1620
(1315)
IN
LP
INC
IND
INE
INF
ING
INH
INI
INB
INS
INSS
ISOLV
(11, 19A4)
NSTOP
TITLE
(1615)
NOPT
ICODE
NTTS
IVEL
I/O file numbers and equation solver used
General input filt number (default 5)
Output file number (default 6)
Initial concentrations
Diffusion coefficients
Mode point bottom elevations (initial)
Node point flow velocities
Settling velocities
Flow depths
Finite element grid geometry data
Boundary conditions
Salinities
New nodal . salinities
0  uses band solver
1  uses frontal solution technique
Job stop and title
0  continue
1  end of job
Job title
Job control parameters, input codes, and
problem options
Type of problem
1  steady state transport problem
2  unsteady transport problem
3  sediment transport problem
4  consolidation problem only
Output control for nonsediment problems
0  standard output
1  compares with analytic solution
calculated in Subroutine EXACT
Number of time steps
Determines initial velocity field, i.e.,
at time step #1 (for unsteady problems
only)
1  velocity components in x and y
directions are set equal to
constants CONXV and CONYV read
in Subroutine VEL
2  each nodal velocity read in from
input file INF
3  velocity computed using user supplied
routine in Subroutine VEL
309
2125 : lELEV
2630 : IDIFl
3135 : IBED
3640 : ISET
4145 : IDEP
4650 : ICONC
5155 : INBC
4660 : IDRY
Elevation of bottom above a given datum
at node points
0  all elevations set equal to 0.0.
1  read each value read in from file
number INE
Initial diffusion coefficient values at
each node
1  and D are set equal to constants
read in Subroutine DISPER
2  nodal diffusion coefficients are read
in from file number I NO
3  diffusion coefficients are calculated
using user supplied procedure
Initial bed profile
0  no sediment present on bed
1  bed profile read in Subroutine ORGBED
Initial settling velocity at each node
1  set to a constant read in Subroutine
SETVEL
2  each settling velocity is read in
from file number ING
3  settling velocities are computed from
model in Subroutine SETVEL
Initial depts of flow at each node
1  set to constant read in Subroutine
DEPTH
2  read in from file number INH
3  computed accordign to user supplied
procedure in Subroutine DEPTHS
Initial suspended sediment concentrations
1  set to constant
2  read in from file number INC
3  computed according to user supplied
procedure in Subroutine CONCIC
Boundary conditions
1  each value read in from file number
1MB
2  computed in Subroutine CONCBC using
user supplied routine
Code to indicate dry node (i.e. negative
flow depth) problem
0  no dry nodes will occur
1  possible dry nodes
5165 : ISS
Code to indicate whether sediment trans
port problem occurs in steady or unsteady
flow
310
0  unsteady flow
1  steady state flow
6670 : ISOUR Code to indicate if local sediment source
of sink is located at any node
0  no source/sink
1  source/sink occurs at one or more
nodes
SET B MESH DATA
These data are read unformatted from file unit INI
CARD B.l
NE Number of elements in system
NP Number of nodes in system
CARD B.2 et.seq.
NOPd.K) Nodal connections read
counterclockwise (8 percard for
quadrilateral element,
6 for triangular element)
CARD B.3 et.seq.
C0RD{J,1) Xcoordinate (meters)
C0RD(J,2) Ycoordinate (meters)
SET C
TRANSIENT PROBLEM INPUT
CARD C.l
TETA
DT
TIM(l)
CARD C.2 et.seq. (8011)
l,etc. : NPMA(I)
for I=1,.„.,NTTS
Transient input
Degree of Implicitness for CrankNicholson
time marching scheme
0  explicit
1  implicit
Time step  sees (should be of the order
6001800 seconds for sediment transport
problems in estuaries)
Starting time  sees
Code to change time step
The value of NPMA at each time determines
if the time step will be changed
0  no change
1  double time step
2  halve time step
311
CARD C.3 et.seq.(80Il)
l,etc. : IFFd.l)
CARD C.4 et.seq.(80Il)
l.etc. : IFF{I,2)
CARD C.5 et.seq.(80Il)
l.etc„ : IVCOD(I)
CARD C.6 et.seq.(80Il)
l.etc. : IDIF(I)
CARD C.7 et.seq.(80Il)
l,etc. : IDEPC(I)
Code for new boundary conditions
Determines if there are new boundary
conditions
0  no change in boundary conditions
1  each value read in from cards
2  computed in Subroutine CONCBC using
user supplied procedure
3  each value read in from file number
INB
Output control
0  no output
1  sedimentation data only
2  concentrations only
3  concentrations and sediment transoort
data ■
New velocities
Same as IVEL but for each time step
0  no new velocities
1  X 8 Y velocities set equal to
constants CONXV and CONYV read in
Subroutine VEL
2  each nodal velocity read in from
input file INF
3  velocity computed using user supplied
routine in Subroutine VEL
New dispersion coefficients
Same as IDIF
0  no new dispersion coefficients
1  D^ and D are set equal to constants
read in Subroutine DISPER
2  nodal dispersion coefficients are
read in from file IND
3  dispersion coefficients are
calculated using dispersion
algorithm
New depths of flow
Same as IDEP
0  no new depths
1  set to constant read in Subroutine
DEPTH
2  read in from file IHN
3  computed according to user supplied
procedure in Subroutine DEPTH
312
CARD C.8 et.seq.(80Il)
l.etc. : ISALC(I)
CARD C.9 et.seq.(80Il)
l.etc. : ISVS(I)
CARD C.IO et.seq.(80Il)
l.etc. : ISORS(I)
CARD C.ll
NHIS
NELH
Mew salinities
0  no new salinities
1  set to a constant read in Subroutine
DEM STY
1  new salinities at specified nodes are
read in Subroutine DENSTY
3  salinities for all nodes are read in
Subroutine DENSTY
Only for sediment problems H0PT=3
New settling velocities. Same as ISET
0  no new settling velocities
1  set to a constant read in Subroutine
SETVEL
2  each settling velocity is read in
from file number JNG
3  settling velocities are computed from
model in Subroutine SETVEL
Code for local sediment source or sink
Same as ISOUR
0  no source/sink
1  source/sink occurs at one or more
nodes
Output control
Number of elements for which time
history will be written
Element numbers
For Steady State Sediment Problems (ISS.NE.O), specify at which nodes and
elements erosion and deposition initially (i.e. first time step) occur
CARD C.12 et.seq.(80Il) Code for nodes
l.etc. : ISTP(I)
for 1=1,..., NP
0  deposition occurs initially
1  erosion occurs initially
CARD C.13 et.seq.(80Il) Code for elements
l.etc. : ISTE(I)
for I=1,...,NE
0  deposition occurs initially
1  erosion occurs initially
SET D WATER AND SEDIMENT PROPERTIES
Read in Subroutine DENSTY
313
CARD D.l
(FIO.5,110) Water parameters
110
1120
TMP
IS
If IS = 0:
CARD D.2 (F10.5)
110 : SW
If IS = 1:
CARD D.2 et.seq.(7F10.5)
110 : SAL(I)
Average water temperature CO
Determines how initial salinities are
read in
0  constant salinity for all nodes
1  salinity for each node is read in
Constant salinity
Value of constant salinity  ppt
Nodal salinity values
rth
Salinity value for lIIL node  ppt
If ISOUR.NE.O, read source/sink term at appropriate nodes
CARD D.3 (4(110, FIO. 5)
110 : IT(J) Node number
1120 : TEMP(J) Local source/sink term  Kg/m^
Reading stops for IT(J) < 0
Read in Subroutine SEDPRP for sediment problems (N0PT=3)
CARD D.4
110
1120
2130
3140
CARD D.5
110
1120
2130
3140
4150
5160
CARD D.6
110
1120
(4F10.5)
Settling velocity
parameters
CRCN=C,
See equations
for
below 
CRCN2=Cp
See equations
for
below 
CRCN3=C^
See equations
for
below 
GAC
Density of sediment mineral
(6F10.5)
AA=Ai
See equations
for
below
AB=Ao
See equations
for
^s
below
AC=Ao
See equations
for
^^s
below
B
See equations
for
<
below
F
See equations
for
Ws
below
AL
See equations
for
^^s
below
(2E10.3)
WSl=W3i
See equations
for
below 
Equivalent sediment particle diameter at
^50 ■
314
CARD D.7
(2F10.5)
110
1120
CARD D.8
110
1120
2130
CARD 0.9
110
1120
EXPNl=ni
EXPN2=n2
(3F10.5)
EXPMl=mT
EXPM2=nip
EXPM3=m3
(2E10.3)
WSK1=K
WSK2=K;
1
See equations for below
See equations for below
See equations for W below
See equations for W below
See equations for W below
See equations for below  m/s
See equations for below  m/s
NOTE: For RANGE I and C < C^ in Range II
= AA*WS1*(SAL)**EXPM1 for C < CRCNl where SAL = salinity
If (SAL < 0.1 ppt)SAL = 0.1 ppt
= AA*WSK1*C**EXPN1*(SAL)**EXPM1 for CRCNl < C < CRCN2
W3 = AB*WSK2*C**EXPN2*(SAL)**EXPM2 for CRCN2 < C < CRCN3
Wg = AC*G*D**2*(GAC/GAW1)*250*(C/CRCN31)**AL*(SAL)**EXPM3/
(18*v*D**1.8) for C > CRCN3
If there is only one W = kc" relationship between C = CRCNl and
the concentration at which hindered settling begins set CRCN3 =
CRCN2 in CARD D.4.
NOTE: For C > C^ in Range II
T=AL0G10((T/T5q)*B*(SAL)**F)**(1./SIG2)
CARD D.IO (2I10, 2F10.5) Properties of new deposits
IIO
1120
2130
3140
NLAYTM
NLAYT
TAUMIN
TAUMAX
Number of layers formed by unconsolidated
new deposits (UND)
Number of layers formed by partially
consolidated new deposits (CND)
mm
max
Parameters characterizing functional relationship between and log^Q
(tgQ) and CT^ at a salinity of 35 ppt.
CARD D.ll
110
1120
(5F10.5)
Al
Sl
See equations for 02 below
See equations for 02 below
315
A2
See
equations for
3140 :
S2
See
equations for
4150 :
CI
See
equations for
CARD D.12
(5F10.5)
iin
A3
See
equations for
1120 :
S3
See
equations for
2130 :
A4
See
equations for
3140 :
S4
See
equations for
4150
C2
See
equations tor
NOTE: For
CI :
a, =
Sl*ug + Al
t >
CI :
^2 =
S2*i;g + A2
C2 :
%0 =
eo.^io^s^*"^
For
60.*10(S3*i:g
C2 :
^50 "
J50
•^50
^50
^50
So
below
below
below
below
below
CARD D.13
+ A4)
et.seq.{4(2F10.5)) Shear strength and dry sediment
density for unconsolidated new deposit layers. NLAYTM
pairs of values are read in starting at the bed
surface and proceeding down to the bottom of the
bottom UND layer.
110
1120
SSM(I)
GADM(I)
Bed shear strength  N/m^
Dry sediment density  kg/rn^
for I = 1,..., NLAYTM
CARD D.14 et.seq.(F10.5)
110
for I
CARD D.15
: TLAYM(I)
1,..., NLAYTM
Thickness of unconsolidated new deposit
layer  m
et.seq. (2F10.5) Shear strength and dry sediment
density for partially consolidated new deposit layers.
NLAYT+1 pairs of values are read starting at the top
of these layers and proceeding downward.
110
1120
for I
CARD D.16
110
1120
2130
SS(I)
GAD(I)
Bed shear strength  N/m^
Dry sediment density  kg/rrr'
1,..., NLAYT+1
et.seq. {3F10. 5) Thickness, and oc values for
each partially consolidated new deposit layer.
TLAY(I) Layer thickness  m
EPSLON(I)  kg/m^/s
ALFA(I) a  dimensionless
315
for I = 1,...,NLAYT
NOTE: The properties read 1n on CARDS D.13D.16 are determined from
laboratory experiments (see Appendix D, Section D.2 for a
description of these experiments). These are the
properties assigned to new deposits if/when deposition occurs
during model simulation or initially if new deposits are
present on top of the original settled bed, as specified in
SET K.
CARD D.17
110
1120
2130
CARD D.18
110
1120
2130
3140
CARD D.19
110
1120
CARD D.20
110
1120
110
1120
(3F10.5) Empirical coefficients in expression for
mean bed density as a function of
consolidation time, T^ .
^c
AP f in Eq. 3.75
ALAMDA p in Eq. 3.75
TCC Time at which the coefficients A and B in
Eq. 3.76 become constants
(4F10.5) Empirical coefficients in expressions for
final mean bed density and corresponding
consolidation time, T^^, .
FMBDO (pJq in Eq. 3.72
AI a in Eq. 3.72
TCINO ^^dcjo 1'" ^<^' 3.73
BI b in Eq. 3.73
(2F10.5) Empirical coefficients in expression for
A (in Eq. 3.76) given by Eq. 3.77.
AQ D (with 9 = A) in Eq. 3.77
BQ F (with 0 = A) in Eq. 3.77
(2F10.5) Empirical coefficients in epression for
B (in Eq. 3.76) given by Eq. 3.77.
EQ D (with 9 = B) in Eq. 3.77
FQ F (with 9=8) in Eq. 3.77
CARD D.21 (2F10.5) Erppirical coefficients in expression for
max
given by Eq. 3.77
PQ
SQ
For NOPT = 2 or 3:
D (with e = z[^^^) in Eq. 3.77
^ ^""'^^ ® = ' max) 1" Eq. 3.77
SET E
INITIAL CONCENTRATION FIELD
The initial concentration at each node must be specified for all unsteady
problems. The type of Input is determined by the value of ICONC.
317
Read from file unit INC
ICONC = 1 Initial concentration set to a constant at all nodes.
CARD E.l {F10.5)
110 : CINT Initial concentration  kg/n?
^CONC = 2 Read in initial concentration for each node.
CARD E.l et.seq.(4(I10,F10.5))
110 : IT{J) Node number
1120 : TEMP{J) Initial concentration  kg/m^
Reading stops for IT(J) < 0
^CONC = 3 Compute initial concentrations at each node using user
supplied model in Subroutine CONCIC.
For NOPT = 3 (Sediment Problems) :
SET F INITIAL BED ELEVATIONS
If IELEV+0, the initial bed elevation, with respect to some datum, at
each node is read in.
Read from file unit INE
CARD F.l et.seq.(8F10.5)
110 : ELEV(I) Bed elevation for node I  m
I = 1,...,NP
For NOPT = 3:
SET G ORIGINAL SETTLED BED PROFILE
Read in only if IBED is not zero. Otherwise the default bed condition
will be a clean bed.
CARD G.l et.seq.(3I5,F10.5) for each element
Element number
Number of layers of original settled
bed for element NN
If NM=0, bed properties are read in for
each element. If NM=i=0, constant values
are read in and used for all elements.
Average density of pore water in original
bed  kg/m3
CARD G.2 et.seq.(2F10.5) Shear strength and dry sediment
density for original settled bed layers. NLA+1
15
: NN
610
■ NLA
1115
NM
1625 :
GWA
318
pairs of values are read in starting at the top layer
and proceeding downward. The first values are for the
top of the original bed.
110 : SSTO(NN,L) Bed shear strength  N/m^
1120 : GADO(NN,L) Dry sediment density  kg/m^
For I = 1,...,NLA+1
CARD G.3 et.seq.(2E10.3) Thickness and value of M for each
settled bed layer.
110 : THICKO(NN,I) Thickness of Ith_ layer  m
1120 : EROCON{NN,I) M value for I th layer  kg/m^/s
For I = 1,...,NLA
Note: CARDS G.l, G.2 and G.3' are repeated for NM=1,...,NE
when NM=0. When NM*0, these cards are read in only
once.
If stationary suspension is present on top of original settled bed, set
NN=10 at the end of the above set (i.e. after CARDS G.l, G.2 and G.3).
For NN=10, read the following cards.
CARD G.4 et.seq.(I10,F10.5) for each element
110 : IT(J) Element number
1120 : TEMP(J) Dry mass per unit area of (soft
unconsolidated sediment) on top of
settled bed  kg/m .
Reading stops when IT(J) < 0
SET H INITIAL DEPTHS OF FLOW
Depths of flow at each node are read in depending on the value of
IDEP. Read from file unit INH
IDEP = 0 All depths set to 1.0 m by default.
IDEP = 1 All nodal depths set to constant.
CARD H.l (Flo. 5) Constant value of depth.
110 : CDEP Depth of flow  m
IDEP = 2 Read node point depths from file INH.
CARD H.2 (4(110, FIO. 5))
110 : IT(J) Node number
1120 : TEMP(J) Depth of flow  m
Stops reading if IT(J) < 0
319
IDEP = 3 Compute depths from user supplied procedure in
Subroutine DEPTH.
SET I INITIAL VELOCITY FIELD
The horizontal velocity components at each node must be specified. The
value of IVEL determines type of input. This input only for unsteady
problems. All reads are from file number INF.
IVEL = 1 Velocities are set to constant values.
Read from file unit INF
CARD I.l (2F10.5)
110 : CONXY Velocity ' component in the xdi recti on 
m/s
1120 : CONYV Velocity component in the ydirection 
m/s
IVEL = 2 Each nodal velocity component read in.
CARD 1.2 (4(2F10.5)) Must be read in order for all NP nodes.
110 : XVEL(J,1) Velocity component at node J in the x
di recti on  m/s
1120 : XVEL{J,2) Velocity component at node J in the y
di recti on  m/s
IVEL = 3 User supplied procedure in Subroutine VEL is
used to calculate nodal velocities.
SET J INITIAL DISPERSION COEFFICIENTS
The form of input is set by the value of IDIF,
IDIF = 1 Dispersion coefficients are set to constant values.
CARD J.l (2F10.5)
110 : CDIFL Longitidinal dispersion coefficient 
m^/s
1120 : CDIFT Transverse dispersion coefficient 
m^/s
IDIE = 2 Dispersion coefficients are read in node by node.
CARD J. 2 et.seq.(3(I5,2F10.5)
15 : IT(J) Node number
615 : TEMP(1,J) Longitudinal dispersion coefficient 
m2/s
1625 : TEMP(2,J) Transverse dispersion coefficient 
".2/s
m'
320
Reading stops for IT(J) < 0
IDIF = 3 Dispersion coefficients D^^^, D^^ , 0^^^ and D
computed analytically using dispersion algorithm.
For NOPT = 3 :
SET K INITIAL SETTLING VELOCITIES
The initial settling velocities at each node point must be read in.
The form of input is determined by the value of I SET.
Read from file unit ING
ISET = 1 All settling velocities are set to constant.
Card K.l {F10.5)
110 : CVSX Settling velocity  m/s
ISET = 2
CARD K.2 et.seq.(4(I10,F10.5))
110 : IT(J) Node number
1120 : TEMP(J) Settling velocity  m/s
Stops reading if IT(J) < 0
ISET = 3 Settling velocity model, for which parameters were
read in SET D, is used to compute each nodal
settling velocity.
SET L BOUNDARY CONDITIONS
For any problem, concentration boundary conditions must be specified
at least at one node. At all external boundaries that have no
concentration specified, the normal diffusive flux is defaulted to zero.
Type of input is determined by value of IN8C. File number for input is INB.
INBC = 1 Read node number and specified boundary condition from
cards.
CARD L.l (3(110, FIO. 5))
110 : IT(J) Node number
1120 : TEMP(J) Specified concentration  kg/m^
For J=1,...,NP
INBC = 2 Concentration computed according to user supplied
procedure in Subroutine CONCBC
321
I NBC = 3 Read node number and specified boundary condition from
tape file 1MB.
CARD L.2
MFIX(J) Equal to 1 for boundary node;
0 for all other nodes
SPEC(J,1) Specified concentration at node J  kg/m"^
For J=1,...,NP
SET M NEW SALINITIES
Type of input is determined by value of ISALC{J), for J=2, . . . ,NTTS.
ISALC(J)=1
CARD M.l (F10.5)
120 : SVI Constant salinity value read in  ppt
ISALC(J)=2 File number for input  INSS
CARD M.2 (3(110, FIO. 5)) Salinities for specified nodes
110 : IT(J) Node number
1120 : TEMP (J) New salinity at node J  ppt
Reading stops where IT(J) < 0.
ISALC(J)=3 File number for input  INSS
CARD M.3 (7F10.5) New salinities for all nodes.
110 : SAL(J) Salinity at Jth node
for J=1,...,NP
DYNAMIC INPUT
The same subroutines that read initial values are used to read changes
in these values during a dynamic run. The input code arrays in SET C
tell the program if any new values should be read in at each time step.
Note that the starting time is time step 1. The order of reading each set
of data is given below. If the code is zero, no input is required. If it
is nonzero, the value of the code will determine the type of input.
322
DESCRIPTION
CODE ARRAY
INPUT CARD SET
Salinities ISALC(J)
Concentration B.C. IFF(J,1)
Depths of flow IDEPC(J)
Velocities IVCOD(J)
Dispersion coefficients IDIF(J)
Settling velocities ISVS(J)
SET M
SET L
SET
SET
SET
SET
I
F
E
J
only for sediment problems
APPENDIX D
DATA COLLECTION AND ANALYSIS PROGRAMS
D.l. Field Data Collection Program
A field data collection program for a sedimentation study should
consist of four principal components: 1) hydrographic survey, 2)
sediment sampling, 3) measurement of suspended sediment concentration,
water temperature and salinity and 4) determination of sediment settling
velocity. The collection program required for modeling the hydrodynamic
regime in an estuary is rather well known, and will not be addressed.
Hydrographic Survey
At least two sonar fathometers and a gammaray transmission
densitometer should be used to measure the depths in the water body to
be modelled. First it is recommended that the entire water body be
surveyed simultaneously using, for example, a 30kHz and a 200kHz
fathometer. The Kelvin Hughes Division of Smiths Industries Ltd. (U.K.)
manufactures a 30kHz fathometer (model MS48) that has an approximate
maximum range and resolution of 1000 m and 10 mm on the 020 range,
respectively. Raytheon (USA) manufacturers a 200kHz fathometer with an
approximate maximum range and resolution of 120 m and 0.5% of depth,
respectively. The 200kHz fathometer has a maximum range that is
approximately one order of magnitude less than that of the 30kHz
instrument because of the greater attenuation of the high frequencies in
water. Raytheon does manufacture a dual 22.5kHz and 200kHz frequency
fathometer. No specifications are available for this model.
323
324
Parker and Kirby (1977) reported that the sedimentwater interfaces
of stationary suspensions in the Severn Estuary. England were detected
by a 200kHz fathometer but not by a 30kHz fathometer. In areas where
stationary suspensions are determined to exist (by comparison of the 30
kHz and 200kHz records), it is recommended that a gammaray
transmission densitometer or a turbidity meter be used to supplement the
depth record obtained with the fathometers. A fathometer alone may not
be capable of detecting the surface of a stationary suspension for the
following reasons: 1) The acoustic detection of a dense suspension
depends on the gradient of the bulk density at the surface of the
suspension and not on the magnitude of the density. In mobile
suspensions and in newly formed stationary suspensions this density
gradient is very small, and in most cases the surfaces of these
suspensions will not be detectable. 2) As the stationary suspension
undergoes consolidation, different parts (i.e. levels) of the suspension
may become detectable to fathometers with different frequencies at
different times, which makes the interpretation of such fathometer
records a difficult and uncertain task (Parker and Kirby, 1977).
A gammaray transmission densitometer obtains in situ measurements
of the sediment bulk density profiles, and thus can be used, in
addition, to determine the thickness of stationary suspensions and the
location (i.e. vertical elevation with respect to geodetic datum) of the
top of the settled bed, at which the bulk density is usually assumed to
be 1300 kg/m . A static cone penetrometer directly measures penetration
resistance and indirectly measures the shear strength of the sediment.
The densitometer has to be calibrated at the beginning and end of survey
operations to determine the relationship between the radiation count
325
rate and sediment density. Calibration is generally performed using
liquids with different densities. The densitometer is penetrated and
retracted in the sediment at a rate of approximately 2 to 3 mm/sec,
during which the radiation count rate and probe penetration depth are
continuously recorded. This system has been repeatedly used from both
ships and submersibles and has the capability of measuring in situ bulk
sediment densities up to 1800 kg/m3. operating in depths up to 3.6 km.
and penetrating one to two meters in cohesive sediment suspensions
(Hirst etal_^, 1975). Harwell, the United Kingdom Atomic Energy
Research Establishment, manufacturers a transmission densitometer that
has a vertical resolution of ± 1 cm with an accuracy of ± 2% in the bulk
density range 10002000 kg/m^ (Parker etal_., 1975).
There are three methods used to fix the boat position during a
hydrographic survey: 1) optical methods. 2) electronic methods and 3)
combined systems. Optical methods include double horizontal sextant
angle resection, single horizontal sextant angle and transit line, and
theodolite intersections from shore. Electronic methods include two
megahertz systems, microwave systems and range and bearing systems,
while combined systems use a theodolite to determine the bearing and a
microwave system to determine the distance. A description of these
three methods is given by Ingham (1975) and Dyer (1979).
Sediment Sampling Using Corers
Before the data collection period begins, at least two 1012 cm
diameter cores should be collected at each sampling station. There are
four types of corers. These are the gravitycorer, pistoncorer,
vibracorer and boxcorer. A gravity corer is lowered close to the ocean
floor and a tripping mechanism is released so that the last part of the
326
descent is in free fall. The core barrel is supposed to penetrate the
sediment, cutting out a cylinder of mud. The barrel is equipped with a
plastic liner which can be slipped out. An orange peel core catcher is
located between the nose piece and the liner to prevent the core from
washing out when the corer is retrieved. In general, gravity corers
have a barrel length of 1 to 2 m.
A piston corer is released and free falls a known distance as soon
as a tripping weight hanging a known distance below the corer senses the
bottom. During penetration of the corer in the sediment the piston
moves up the core liner. This action permits the hydrostatic pressure
head of the water column to aid the corer in penetrating the sediment
and removing the water from the barrel to reduce the resistance to the
core as it enters the liner. Cores in excess of 20 m have been obtained
from soft sediments.
A vibrator mechanism is used to drive a vibracorer into sediment.
In general, this type of corer is heavy and requires a large vessel for
operation. However, a lighweight pneumatic corer has recently been
designed and field tested at the U.S. Army Corps of Engineers Coastal
Engineering Research Center (CERC) (Fuller and Meisburger, 1982). This
system can be used from relatively small vessels for obtaining 5.0 cm
diameter cores of unconsolidated sediments from 0.6 to 2.4 m in length.
A boxcorer is used to obtain large cores (up to 50 cm in height,
30 cm in length and 20 cm in width) of sediment when it is essential
that the internal fabric of the sediment not be disturbed. Boxcorers
consist of an open steel box that is driven vertically into the sediment
until the top of the box rests on the surface of the sediment. The open
end of the box is covered by a blade which cuts through the sediment.
327
This type of corer has been used to obtain excellent cores of estuarine
sediment (Dyer 1979).
Measurement of Suspen ded Concentration. Salinity and Temperature
The first item that must be considered is the time period over
which data will be collected for eventual use in the model. This will
be contingent upon the desired results from the modeling effort. In
tidal water bodies, data should preferably be collected over a minimum
of 15 hours (assuming the tide is semidiurnal) over three different
tidal cycles: spring, mean and neap. It would be more desirable to
have the data collection period span at least one week, starting, for
example, on a spring tide and finishing at the subsequent neap tide.
The next consideration is the number of data sampling stations and where
they should be located in order to adequately monitor the spatial
variations of the concentration of suspended sediment. Stations must be
located at all exterior water boundaries (crosssections) of the
estuarial system to be modeled. The width of the boundary crosssection
and the lateral variability of the depth should be considered when
deciding upon the minimum number of stations to be located laterally
across such a boundary. For example, stations would definitely be
located at predominant features such as navigation channels. Additional
stations must be located at all interior confluences and bifurcations,
and at as many other interior locations as possible. It is recognized
that the length of the data collection period and the number of stations
are often less than desired due to economic and logistical
considerations.
At each station the location of the top of the sediment bed with
respect to a geodetic or tidal datum must be determined using the
328
previously described surveying methods. It is recommended the water
temperature, electrical conductivity (or salinity) and concentration of
suspended sediment be measured at least once every onehalf hour for the
duration of the collection period at each of the sampling stations.
These measurements should preferably be made at a minimum of three
depths over the vertical: onehalf meter below the water surface, mid
depth and onehalf meter above the bottom (i.e. top of sediment bed).
For locations where the water depth is greater than about 3 to 4 m,
measurements should be made at additional depths over the vertical.
Both the measurement and analysis of water temperature and electrical
conductivity data are discussed by Dyer (1979). A description of
various filtration procedures for determining the suspension
concentration gravi metrically is given by Dyer (1979).
There are three general methods used to measure the suspension
concentration: 1) water sampling, 2) optical methods and 3) gammaray
densitometer measurements.
Water bottles and shipboard pumps are the two most common water
sampling devices. The NIO bottle has capacities from 1.25 to 7.1
liters. Other water bottles, such as the Van Dorn bottle, have
capacities of up to 10 liters or more. The NIO bottle consists of a PVC
tube open at both ends with hemispherical bings on spring loaded arms
which close each end. Most water bottles are closed by dropping a brass
messenger down the support wire (Dyer, 1979).
Shipboard pumps are used to pump water samples up to the vessel
through an intake tube mounted on an instrument package. It is
recommended that in situ separation of the water and sediment be
performed on the vessel using the filter method (van Rijn, 1979).
329
Instruments for optical determination of the concentration of
suspended sediment include the transmissometer, nephelometer and the
Secchi disc. Nephelometers are not very practical for use in
estuaries since they are sensitive to very low concentrations only.
Secchi discs can be used to estimate surface values only.
Transmissometers, or electrooptical turbidity meters, have been used
successfully to measure vertical turbidity profiles in, among others,
the Severn, Maas, James and Rappahannock estuaries and in Upper
Chesapeake Bay (Kirby and Parker, 1977; Nichols ^al_., 1979). These
meters can be used to detect both mobile and stationary suspensions as
their operating range, in general, is 0.2525 g/1 . They have a rapid
response time (100 Hz) which allows profiles in 30 m depths to be taken
in 1520 seconds. Partech (U.K.) is one manufacturer of optical
turbidity meters.
Both Kirby and Parker (1977) and Nichols etaj_. (1979) used
instrument arrays on which were mounted a gammaray transmission
densitometer, at least one electrooptical turbidity meter, an
electromagnetic current meter, a pressure transducer and a water
temperature and conductivity probe. In general, the optical turbidity
meters would be used to record the concentration profile for suspensions
up to 25 g/1 and the transmission densitometer used for denser
suspensions.
Determination of Sediment Settling Velocity
An appropriate method to measure settling velocities is by using an
instrument similar to the sampling tube developed by Owen (1971), in
which undisturbed samples of suspended sediments are collected in situ
in their natural state. The settling velocities of the aggregates are
330
determined immediately thereafter through use of a bottom withdrawal
sedimentation test. Allersma (1980) gives a detailed description of an
I'n situ suspended sediment sampler.
D.2 Laboratory Sediment Testing Program
It is recommended that the following physiochemical sediment and
fluid properties be determined using the collected sediment cores.
Properties of Undisturbed Sediment Cores
The gammaray densitometer may be used to determine the bulk
sediment density profile in the undisturbed cores still in the liner
tubes as soon after the cores are obtained as possible. A description
of this procedure is given by Whitmarsh (1971) and Kirby and Parker
(1974). If this instrument is not available, the freezedrying
procedure used by Parchure (1980) and Dixit (1982) or the pumping method
used by Thorn and Parsons (1977) may be used to determine the bulk
density profile. The pumping method consists of removing by suction a
thin layer, e.g. 3 cm, from the top of the core. This procedure is
repeated, layer by layer, and each layer is analyzed to determine the
mean bulk density.
Properties of Original Settled Bed
The bulk density and bed shear strength profiles and the erosion
rate constant for each layer need to be determined for the cores. The
number of layers and the thickness of each are determined from the
nature of the bed shear strength profile. The erosion rate constant for
each layer and the shear strength profile can be determined, for
example, in the rotating cylinder erodibility testing apparatus
described by Sargunam et al_. (1973). In order to use this apparatus.
331
the core sample must be trimmed. The portion of each core that is
sufficiently consolidated such that it can be trimmed and tested in the
erosion apparatus is defined to be the settled bed. The thickness of
this portion defines the location of the top of the settled bed. Soft,
unconsolidated portions of each core are assumed to be new deposits.
Properties of New Deposits
For cores with soft, unconsolidated or partially consolidated
sediment on top of the settled portion, the following method may be used
to estimate the erosional, depositional and consolidation
characteristics of such new deposits. The new deposit samples from the
cores at all the stations should be mixed and subjected to laboratory
erosion, deposition and consolidation tests (described by Parchure
(1980), Mehta and Partheniades (1973), in Sections 3.4 and 3.6 and
below) to determine: the settling velocity as a function of suspension
concentration and salinity; the minimum and maximum depositional shear
stresses c^^:^^ and z^^^^; variation of tgg and with the bed shear
stress, v^; the number of characteristic stationary suspension layers,
and the thickness, dry sediment density and shear strength of each
layer; the number of characteristic partially consolidated new deposit
layers, and the thickness, dry sediment density, shear strength and the
resuspension parameters and a of each layer; the variation of 9^ and
Tdc„ with C^; variation of p(z) with T^^; variation of the bed shear
strength, t^, with p. The variation of the bed density and shear
strength profiles with salinity can be determined by performing the
erosion tests at several salinities between 0 and 35 ppt. A brief
description of laboratory tests which can be conducted in order to
determine the above mentioned consolidation parameters is given next.
332
Laboratory tests to determine the consolidation characteristics of
a cohesive sediment bed involve the measurement of the bed density
profile. Various methods have been used for this purpose. Been and
Sills (1931) measured the density profile of a clayey soil using a non
destructive Xray technique. Methods which involve the destruction of
the soil include, among others, the freezedrying procedure used by
Parchure (1980) and Dixit (1982), the pumping, or layerbylayer
sampling method used by Thorn and Parsons (1977), and the use of
specially designed apparatuses (Parchure, 1980). The latter consists of
a 183 cm high, 30 cm diameter PVC cylinder, a bottom plate, and ten 1.27
cm diameter plastic tubes ranging from 1.27 to 12.7 cm in height glued
to the bottom plate, concentric to the PVC cylinder. These cylinders
are filled with a sediment suspension of known concentration, and the
sediment is allowed to settle under quiescent conditions for a specified
consolidation time. Following the procedure described by Parchure
(1980), the bulk density profile can be determined.
The following parameters should be varied systematically in the
laboratory tests in order to determine their effect on the rate of
consolidation:
(a) Consolidation time  it is recommended that the time allowed for
the bed to consolidate before the density profile is measured be
varied logarithmically from 0 to 720 hours (1 month).
(b) Initial conditions  the initial suspension concentration, which
determines the thickness and density of the initial bed.
(c) Salinity  the salinity of the water should be varied from 0 (tap
water) to 35 ppt.
333
(d) Overburden  it is important that the effect of discretized to
continuous additions of varying amounts of sediment (overburden) on
top of the initial bed be determined in order to evaluate the
effect of such overburden pressures on the consolidation rate of
the lower bed layers.
The relationship between p and needs to be determined as well
for the collected sediment samples. Both the bed shear strength and
density profiles may be determined using the methodology described by
Mehta et al. (1982a). These profiles can then be used to establish an
empirical relationship between and p.
Fluid Composition
The pH, total salt concentration, and concentrations of ions such
as Na"*", Ca^"^, Mg^"^, K"^, Fe^"^ and CI" should be determined for both the
pore fluid in the consolidated bed portion of one core and a sample of
the suspending fluid.
Composition and Cation Exchange Capacity of the Sediment
The sediment contained in the consolidated bed portion of one core
from each collection station should be thoroughly mixed so that a
spatially homogeneous sample is obtained. A standard hydrometer
analysis must be conducted on each soprepared sample to determine the
sediment particle size distribution and thereby the percentage by weight
of clay, silt and fine to coarse sand in each sample. In preparing the
samples for this analysis, the sediment must not be initially airdried
(to obtain the dry weight of the material used in the test), as it has
been found that dried sediment will not completely redisperse when the
dispersing agent is added (Krone, 1962). For this reason, the total dry
weight of the sample must be obtained after the test by evaporating off
334
all the water in an oven set at approximately 50° C. The percentage of
weight of organic matter should be determined through use of a method
such as the WalkleyBlack test (Allison, 1965). In addition, it is
recommended that Xray diffraction analysis of the bulk sample, and < 2
m unglycolated and glycolated portions be conducted in order to
determine the predominant clay and nonclay mineral constituents. The
cation exchange capacity must be determined for each sample.
335
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BIOGRAPHICAL SKETCH
Earl Joseph Hayter was born November 19, 1954, in Coral Gables,
Florida. At the age of 9 his family moved to Apopka, Florida, where he
completed his primary and secondary education and graduated from Lyman
High School in 1972. He attended Florida Technological University from
September 1972 to June 1974. In September 1974 he transferred to
Florida Institute of Technology and majored in physical oceanography.
He graduated from F.I.T. in June 1975 with a Bachelor of Science degree
in oceanography. In September 1976 he enrolled as a graduate student in
the Coastal and Oceanographic Engineering Department at the University
of Florida. He graduated from the University of Florida in March 1979
with a Master of Science degree in coastal and oceanography
engineering. From September 1977 to May 1979 he worked for Suboceanic
Consultants, Inc., in Naples, Florida.
He reentered graduate school at the University of Florida in June
1979 in the Department of Civil Engineering to work toward the Doctor of
Philosophy degree. On June 21, 1980, he married Janet Griffen. He
worked as a graduate research assistant in the Coastal and Oceanographic
Engineering Department during the periods September 1976  September
1977 and June 1979  September 1981. From October 1981 to September
1983, he was supported by the Water Resources Division of the U.S.
Geological Survey under their thesis support program.
349
I certify that I have read this study and that in my
opinion 1t conforms to acceptable standards of scholarly
presentation and is fully adequate in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Bent A. Christensen, C hai rman
Professor of Civil Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
/A sh^*W^J. Mehta, C o" C h a i r ma h
Associate Professor of Coastal
and Oceanographic Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Barry ^ Benedict
Professor of Civil Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
WAjL^ ex rQ.r^!l^^4^
/J^mes ~. Eades
Associate Professor of Geology
Tin's dissertation v/as submitted to the Dean of the College of
Engineering and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
December, 1983
Dean, College of Engineering
Dean, Graduate School